349
ISSN 1464-8997
Geometry & Topology Monographs
Volume 2: Proceedings of the Kirbyfest
Pages 349–406
Structure of the mapping class groups of surfaces:
a survey and a prospect
Shigeyuki Morita
Abstract In this paper, we survey recent works on the structure of the
mapping class groups of surfaces mainly from the point of view of topology. We then discuss several possible directions for future research. These
include the relation between the structure of the mapping class group and
invariants of 3–manifolds, the unstable cohomology of the moduli space of
curves and Faber’s conjecture, cokernel of the Johnson homomorphisms and
the Galois as well as other new obstructions, cohomology of certain infinite
dimensional Lie algebra and characteristic classes of outer automorphism
groups of free groups and the secondary characteristic classes of surface
bundles. We give some experimental results concerning each of them and,
partly based on them, we formulate several conjectures and problems.
AMS Classification 57R20, 32G15; 14H10, 57N05, 55R40,57M99
Keywords Mapping class group, Torelli group, Johnson homomorphism,
moduli space of curves
This paper is dedicated to Robion C Kirby on the occasion of his 60th birthday.
1
Introduction
Let Σg be a closed oriented surface of genus g ≥ 2 and let Mg be its mapping
class group. This is the group consisting of path components of Diff + Σg , which
is the group of orientation preserving diffeomorphisms of Σg . Mg acts on the
Teichmüller space Tg of Σg properly discontinuously and the quotient space
Mg = Tg /Mg is the (coarse) moduli space of curves of genus g . Tg is known
to be homeomorphic to R6g−6 . Hence we have a natural isomorphism
H ∗ (Mg ; Q) ∼
= H ∗ (Mg ; Q).
On the other hand, by a theorem of Earle–Eells [20], the identity component
of Diff + Σg is contractible for g ≥ 2 so that the classifying space BDiff + Σg
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350
Shigeyuki Morita
is an Eilenberg–MacLane space K(Mg , 1). Therefore we have also a natural
isomorphism
H ∗ (BDiff + Σg ) ∼
= H ∗ (Mg ).
Thus the mapping class group serves as the orbifold fundamental group of the
moduli space Mg and at the same time it plays the role of the universal monodromy group for oriented Σg –bundles. Any cohomology class of the mapping
class group can be considered as a characteristic class of oriented surface bundles and, over the rationals, it can also be identified as a cohomology class of
the moduli space.
The Teichmüller space Tg and the moduli space Mg are important objects
primarily in complex analysis and algebraic geometry. Many important results concerning these two spaces have been obtained following the fundamental
works of Ahlfors, Bers and Mumford. Because of the limitation of our knowledge, we only mention here a survey paper of Hain and Looijenga [43] for recent
works on Mg , mainly from the viewpoint of algebraic geometry, and a book by
Harris and Morrison [53] for basic facts as well as more advanced results.
From a topological point of view, fundamental works of Harer [46, 47] on the
homology of the mapping class group and also of Johnson (see [63]) on the
structure of the Torelli group, both in early 80’s, paved the way towards modern
topological studies of Mg and Mg . Here the Torelli group, denoted by Ig , is
the subgroup of Mg consisting of those elements which act on the homology of
Σg trivially.
Slightly later, the author began a study of the classifying space BDiff + Σg of
surface bundles which also belongs to topology. The intimate relationship between three universal spaces, Tg , Mg and BDiff + Σg described above, imply
that there should exist various interactions among the studies of these spaces
which are peculiar to various branches of mathematics including the ones mentioned above. Although it is not always easy to understand mutual viewpoints,
we believe that doing so will enhance individual understanding of these spaces.
In this paper, we would like to survey some aspects of recent topological study
of the mapping class group as well as the moduli space. More precisely, we
focus on a study of the mapping class group which is related to the structure of
the Torelli group Ig together with a natural action of the Siegel modular group
Sp(2g, Z) on some graded modules associated with the lower (as well as other)
central series of Ig . Here it turns out that explicit descriptions of Sp–invariant
tensors of various Sp–modules using classical symplectic representation theory,
along the lines of Kontsevich’s fundamental works in [85, 86], and also Hain’s
recent work [41] on Ig using mixed Hodge structures can play very important
Geometry and Topology Monographs, Volume 2 (1999)
Structure of the mapping class groups of surfaces
351
roles. These two points will be reviewed in section 4 and section 5, respectively.
In the final section (section 6), we describe several experimental results, with
sketches of proofs, by which we would like to propose some possible directions
for future research.
This article can be considered as a continuation of our earlier papers [113, 114,
117].
Acknowledgements We would like to express our hearty thanks to R Hain,
N Kawazumi and H Nakamura for many enlightening discussions and helpful
information. We also would like to thank C Faber, S Garoufalidis, M Kontsevich, J Levine, E Looijenga, M Matsumoto, J Murakami and K Vogtmann
for helpful discussions and communications. Some of the explicit computations
described in section 6 were done by using Mathematica. It is a pleasure to
thank M Shishikura for help in handling Mathematica.
2
Mg as an extension of the Siegel modular group
by the Torelli group
Let us simply write H for H1 (Σg , Z). We have the intersection pairing
µ : H ⊗ H−→Z
which is a non-degenerate skew symmetric bilinear form on H . The natural
action of Mg on H , which preserves this pairing, induces the classical representation
ρ0 : Mg −→ Aut H.
If we fix a symplectic basis of H , then Aut H can be identified with the Siegel
modular group Sp(2g, Z) so that we can write
ρ0 : Mg −→Sp(2g, Z).
The Torelli group, denoted by Ig , is defined to be the kernel of ρ0 . Thus we
have the following basic extension of three important groups
1−→Ig −→Mg −→Sp(2g, Z)−→1.
(1)
Associated to each of these groups, we have various moduli spaces. Namely the
(coarse) moduli space Mg of genus g curves for Mg , the moduli space Ag of
principally polarized abelian varieties for Sp(2g, Z) and the Torelli space Tg
for Ig . Here the Torelli space is defined to be the quotient of the Teichmüller
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Shigeyuki Morita
space Tg by the natural action of Ig on it. Since Ig is known to be torsion
free, Tg is a complex manifold. We have holomorphic mappings between these
moduli spaces
Tg −→Mg −→Ag
where the first map is an infinite ramified covering and the second map is
injective by the theorem of Torelli.
By virtue of the above facts, we can investigate the structure of Mg (or that
of Mg ) by combining individual study of Ig and Sp(2g, Z) (or Tg and Ag )
together with some additional investigation of the action of Sp(2g, Z) on the
structure of the Torelli group or Torelli space. Here it turns out that the symplectic representation theory can play a crucial role. However, before reviewing
them, let us first recall the fundamental works of D Johnson on the structure of
Ig very briefly (see [63] for details) because it is the starting point of the above
method.
Johnson proved in [62] that Ig is finitely generated for all g ≥ 3 by constructing
explicit generators for it. Before this work, a homomorphism
τ : Ig −→Λ3 H/H
was introduced in [61] which generalized an earlier work of Sullivan [136] extensively and is now called the Johnson homomorphism. Here Λ3 H denotes the
third exterior power of H and H is considered as a natural submodule of Λ3 H
by the injection
H 3 u 7−→ u ∧ ω0 ∈ Λ3 H
P
where ω0 ∈ Λ2 H is the symplectic class (in homology) defined as ω0 = i xi ∧yi
for any symplectic basis xi , yi (i = 1, · · · , g) of H .
Let Kg ⊂ Mg be the subgroup of Mg generated by all Dehn twists along
separating simple closed curves on Σg . It is a normal subgroup of Mg and is
contained in the Torelli group Ig . In [64], Johnson proved that Kg is exactly
equal to Ker τ so that we have an exact sequence
τ
1−→Kg −→Ig −→Λ3 H/H−→1.
(2)
Finally in [65], he determined the abelianization of Ig for g ≥ 3 in terms of certain combination of τ and the totality of the Birman–Craggs homomorphisms
defined in [12]. The target of the latter homomorphisms are Z/2 so that the
first rational homology group of Ig (or more precisely, the abelianization of Ig
modulo 2 torsions) is given simply by τ . Namely we have an isomorphism
τ : H1 (Ig ; Q) ∼
= Λ3 HQ /HQ
where HQ = H ⊗ Q.
Geometry and Topology Monographs, Volume 2 (1999)
Structure of the mapping class groups of surfaces
353
Problem 2.1 Determine whether the Torelli group Ig (g ≥ 3) is finitely presentable or not. If the answer is yes, give an explicit finite presentation of it.
It should be mentioned here that Hain [41] proved that the Torelli Lie algebra
tg , which is the Malcev Lie algebra of Ig , is finitely presentable for all g ≥ 3.
Moreover he gave an explicit finite presentation of tg for any g ≥ 6 which turns
out to be very simple, namely there arise only quadratic relations. Here a result
of Kabanov [68] played an important role in bounding the degrees of relations.
More detailed description of this work as well as related materials will be given
in section 5.
On the other hand, in the case of g = 2, Mess [99] proved that I2 = K2 is an
infinitely generated free group. Thus we can ask
Problem 2.2 (i) Determine whether the group Kg is finitely generated or
not for g ≥ 3.
(ii) Determine the abelianization H1 (Kg ) of Kg .
We mention that Kg is far from being a free group for g ≥ 3. This is almost
clear because it is easy to construct subgroups of Kg which are free abelian
groups of high ranks by making use of Dehn twists along mutually disjoint
separating simple closed curves on Σg . More strongly, we can show, roughly as
follows, that the cohomological dimension of Kg will become arbitrarily large
if we take the genus g sufficiently large. Let
τg (2) : Kg −→hg (2)
be the second Johnson homomorphism given in [115, 118] (see section 5 below
for notation). Then it can be shown that the associated homomorphism
τg (2)∗ : H ∗ (hg (2))−→H ∗ (Kg )
is non-trivial by evaluating cohomology classes coming from H ∗ (hg (2)), under
the homomorphism τg (2)∗ , on abelian cycles of Kg which are supported in the
above free abelian subgroups.
In section 6.6, we will consider the cohomological structure of the group Kg from
a hopefully deeper point of view which is related to the secondary characteristic
classes of surface bundles introduced in [119].
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3
Shigeyuki Morita
The stable cohomology of Mg and the stable homotopy type of Mg
Let π : Cg →Mg be the universal family of stable curves over the Deligne–
Mumford compactification of the moduli space Mg . In [122], Mumford defined
certain classes
κi ∈ Ai (Mg )
in the Chow algebra (with coefficients in Q) of the moduli space Mg by setting
κi = π∗ (c1 (ω)i+1 ) where ω denotes the relative dualizing sheaf of the morphism
π . On the other hand, in [107] the author independently defined certain integral
cohomology classes
ei ∈ H 2i (Mg ; Z)
of the mapping class group Mg by setting ei = π∗ (ei+1 ) where
π : EDiff + Σg → BDiff + Σg
is the universal oriented Σg –bundle and e ∈ H 2 (EDiff + Σg ; Z) is the Euler class
of the relative tangent bundle of π . As was mentioned in section 1, there exists a
natural isomorphism H ∗ (Mg ; Q) ∼
= H ∗ (Mg ; Q) and it follows immediately from
i+1
the definitions that ei = (−1) κi as an element of these rational cohomology
groups. The difference in signs comes from the fact that Mumford uses the
first Chern class of the relative dualizing sheaf of π while our definition uses
the Euler class of the relative tangent bundle. These classes κi , ei are called
tautological classes or Mumford–Morita–Miller classes.
In this paper, we use our notation ei to emphasize that we consider it as an
integral cohomology class of the mapping class group rather than an element of
the Chow algebra of the moduli space. A recent work of Kawazumi and Uemura
in [78] shows that the integral class ei can play an interesting role in a study
of certain cohomological properties of finite subgroups of Mg .
Let
Φ : Q[e1 , e2 , · · · ]−→ lim H ∗ (Mg ; Q)
g→∞
(3)
be the natural homomorphism from the polynomial algebra generated by ei
into the stable cohomology group of the mapping class group which exists by
virtue of a fundamental result of Harer [47]. It was proved by Miller [102] and
the author [108], independently, that the homomorphism Φ is injective and we
have the following well known conjecture (see Mumford [122]).
Geometry and Topology Monographs, Volume 2 (1999)
Structure of the mapping class groups of surfaces
355
Conjecture 3.1 The homomorphism Φ is an isomorphism so that
lim H ∗ (Mg ; Q) ∼
= Q[e1 , e2 , · · · ].
g→∞
We would like to mention here a few pieces of evidence which support the above
conjecture. First of all, Harer’s explicit computations in [46, 49, 51] verify the
conjecture in low degrees. See also [4] for more recent development. Secondly,
Kawazumi has shown in [72] (see also [71, 73]) that the Mumford–Morita–Miller
classes occur naturally in his algebraic model of the cohomology of the moduli
space which is constructed in the framework of the complex analytic Gel’fand–
Fuks cohomology theory, whereas no other classes can be obtained in this way.
Thirdly, in [76, 77] Kawazumi and the author showed that the image of the
natural homomorphism
H ∗ (H1 (Ig ); Q)Sp −→H ∗ (Mg ; Q)
is exactly equal to the subalgebra generated by the classes ei (see section 6.4 for
more detailed survey of related works). Here Sp stands for Sp(2g, Z). Finally,
as is explained in a survey paper by Hain and Looijenga [43] and also in our
paper [76], a combination of this result with Hain’s fundamental work in [41] via
Looijenga’s idea to use Pikaart’s purity theorem in [133] implies that there are
no new classes in the continuous cohomology of Mg , with respect to a certain
natural filtration on it, in the stable range.
Now there seems to be a rather canonical way of realizing the homomorphism
Φ of (3) at the space level. To describe this, we first recall the cohomological
nature of the classical representation ρ0 : Mg →Sp(2g, Z). The Siegel modular
group Sp(2g, Z) is a discrete subgroup of Sp(2g, R) and the maximal compact
subgroup of the latter group is isomorphic to the unitary group U (g). Hence
there exists a universal g –dimensional complex vector bundle on the classifying
space of Sp(2g, Z). Let η be the pull back, under ρ0 , of this bundle to the
classifying space of Mg . As was explained in [7] (see also [108]), the dual
bundle η ∗ can be identified, on each family π : E→X of Riemann surfaces, as
follows. Namely it is the vector bundle over the base space X whose fiber on
x ∈ X is the space of holomorphic differentials on the Riemann surface Ex .
In the above paper, Atiyah used the Grothendieck Riemann–Roch theorem to
deduce the relation
e1 = 12c1 (η ∗ ).
If we apply the above procedure to the universal family Cg →Mg , then we
obtain a complex vector bundle η ∗ (in the orbifold sense) over Mg (in fact, more
Geometry and Topology Monographs, Volume 2 (1999)
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Shigeyuki Morita
generally, over the Deligne–Mumford compactification Mg ) which is called the
Hodge bundle. In [122], Mumford applied the Grothendieck Riemann–Roch
theorem to the morphism Cg →Mg and obtained an identity, in the Chow
algebra A∗ (Mg ), which expresses the Chern classes of the Hodge bundle in
terms of the tautological classes κ2i−1 with odd indices together with some
canonical classes coming from the boundary. From this identity, we can deduce
the relations
2i
e2i−1 =
s2i−1 (η ∗ ) (i = 1, 2, · · · )
(4)
B2i
in the rational cohomology of Mg . Here B2i denotes the 2i-th Bernoulli number
si (η ∗ ) is the characteristic class of η ∗ corresponding to the formal sum
P and
i
j tj (sometimes called the i-th Newton class). We have also obtained the
above relations in [107] by applying the Atiyah–Singer index theorem [9] for
families of elliptic operators, along the lines of Atiyah’s argument in [7]. Since
η ∗ is flat as a real vector bundle, all of its Pontrjagin classes vanish so that we
can conclude that the Chern classes of η ∗ can be expressed entirely in terms of
the classes e2i−1 . Thus we can say that the totality of the classes e2i−1 of odd
indices is equivalent to the total Chern class of the Hodge bundle which comes
from the Siegel modular group.
Although the rational cohomology of Mg and Mg are canonically isomorphic
to each other, there seems to be a big difference between the torsion cohomology
of them. To be more precise, let
BDiff + Σg = K(Mg , 1)−→Mg
(g ≥ 2)
be the natural mapping which is uniquely defined up to homotopy, where the
equality above is due to a result of Earle and Eells [20] as was already mentioned
in the introduction. As is well known (see eg [46]), Mg is perfect for all g ≥ 3
so that we can apply Quillen’s plus construction on K(Mg , 1) to obtain a
simply connected space K(Mg , 1)+ which has the same homology as that of
Mg . It is known that the moduli space Mg is simply connected. Hence, by the
universal property of the plus construction, the above mapping factors through
a mapping
K(Mg , 1)+ −→Mg .
Problem 3.2 Study the homotopy theoretical properties of the above mapping K(Mg , 1)+ −→Mg . In particular, what is its homotopy fiber ?
The classical representation ρ0 : Mg →Sp(2g, Z) induces a mapping
K(Mg , 1)+ −→K(Sp(2g, Z), 1)+
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Structure of the mapping class groups of surfaces
because Sp(2g, Z) is also perfect for g ≥ 3. Homotopy theoretical properties
of this map (or rather its direct limit as g → ∞) have been studied by many
authors and they produced interesting implications on the torsion cohomology
of Mg (see [15, 16, 36, 138] as well as their references). A final result along these
lines was obtained by Tillmann. This says that K(M∞ , 1)+ is an infinite loop
space and the natural map K(M∞ , 1)+ →K(Sp(2∞, Z), 1)+ is that of infinite
loop spaces (see [138] for details). See also [104] for a different feature of the
above map, [142] for a homotopy theoretical implication of Conjecture 3.1 and
[128] for the etale homotopy type of the moduli spaces.
Let Fg be the homotopy fiber of the above mapping (5). Then, we have a map
Tg −→Fg .
Using the fact that any class ei is primitive with respect to Miller’s loop space
structure on K(M∞ , 1)+ , it is easy to see that the natural homomorphism
Q[e2 , e4 , · · · ]−→H ∗ (Fg ; Q)
is injective in a certain stable range and we can ask how these cohomology
classes behave on the Torelli space.
We would like to show that the classes e2i of even indices are closely related
to the Pontrjagin classes of the moduli space Mg and also of the Torelli space
Tg . To see this, recall that Tg is a complex manifold and Mg is nearly a
complex manifold of dimension 3g − 3. More precisely, as is well known it has
f g which is a complex manifold and we can write
a finite ramified covering M
f g /G where G is a suitable finite group acting holomorphically on M
fg .
Mg = M
Hence we have the Chern classes
f g ; Z)
ci ∈ H 2i (M
(i = 1, 2, · · · )
f g which is invariant under the action of G. Hence
of the tangent bundle of M
we have the rational cohomology classes
coi ∈ H 2i (Mg ; Q)
f g . We may call them
which is easily seen to be independent of the choice of M
orbifold Chern classes of the moduli space. To identify these classes, we use the
Grothendieck Riemann–Roch theorem applied to the morphism π : Cg →Mg
π∗ (ch(ξ)T d(ω ∗ )) = ch(π! (ξ))
where ω ∗ denotes the relative tangent bundle (in the orbifold sense) of π and ξ
is a vector bundle over Cg . If we take ξ to be the relative cotangent bundle ω
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Shigeyuki Morita
as in [122], then we obtain the relations (4) above. Instead of this, let us take
ξ to be ω ∗ . Since π! (ω ∗ ) = −T Mg by the Kodaira–Spencer theory, we have
cho (Mg ) = − π∗ ch(ω ∗ )T d(ω ∗ )
e
= − π∗ exp e
1 − exp−e
∞
n
X
1
Bk 2k o
1
= − π∗ 1 + e + · · · + en + · · · 1 + e +
(−1)k−1
e
n!
2
(2k)!
k=1
where e ∈
H 2 (C
g ; Q)
denotes the Euler class of
ω∗ .
From this, we can conclude
n 1
1
1
1
B1
1
B2
+
· +
+
·−
(2k)! (2k − 1)! 2 (2k − 2)! 2
(2k − 4)!
4!
o
1
B
B
k−1
k
+ · · · + · (−1)k
+ (−1)k−1
e2k−1
2
(2k − 2)!
(2k)!
n
1
1
1
1
B1
1
B2
o
s2k (Mg ) = −
+
· +
+
·−
(2k + 1)! (2k)! 2 (2k − 1)! 2
(2k − 3)!
4!
o
1
Bk−1
Bk
+ · · · + · (−1)k
+ (−1)k−1
e2k .
6
(2k − 2)!
(2k)!
so2k−1 (Mg ) = −
The first few classes are given by
so1 (Mg ) = −
13
e1 ,
12
1
so2 (Mg ) = − e2 ,
2
so3 (Mg ) = −
119
e3 .
720
Thus the orbifold Chern classes of Mg turn out to be, in some sense, independent of g . The pull back of these classes to the Torelli space Tg are equal to
the (genuine) Chern classes of it because Tg is a complex manifold. Since the
pull back of e2i−1 to Tg vanishes for all i, we can conclude that s2i−1 (Tg ) = 0
and only the classes s2i (Tg ) may remain to be non-trivial. As is well known,
these classes are equivalent to the Pontrjagin classes of Tg as a differentiable
manifold.
In view of the above facts, it may be said that the classifying map Mg →BU (3g−
3) of the holomorphic tangent bundle of Mg would realize the conjectural
isomorphism (3) at the space level (rigorously speaking, we have to use some
finite covering of Mg ). Alternatively we could use the map
Mg →Ag × BSO(6g − 6)
where the second factor is the classifying map of the tangent bundle of Mg
as a real vector bundle. In short, we can say that the odd classes e2i−1 serve
Geometry and Topology Monographs, Volume 2 (1999)
Structure of the mapping class groups of surfaces
359
as Chern classes of the Hodge bundle while the even classes e2i embody the
orbifold Pontrjagin classes of the moduli space.
According to Looijenga [91], the Deligne–Mumford compactification Mg can
also be described as a finite quotient of some compact complex manifold. Hence
we have its orbifold Chern classes as well as orbifold Pontrjagin classes. On
the other hand, since Mg is a rational homology manifold, its combinatorial
Pontrjagin classes in the sense of Thom are defined.
Problem 3.3 Study the relations between orbifold Chern classes, orbifold
Pontrjagin classes and Thom’s combinatorial Pontrjagin classes of Mg . In
particular, study the relation between the corresponding charateristic numbers.
If we look at the basic extension (1) given in section 2, keeping in mind the
above discussions together with the Borel vanishing theorem given in [13, 14]
concerning the triviality of twisted cohomology of Sp(2g, Z) with coefficients
in non-trivial algebraic representations of Sp(2g, Q), we arrive at the following
conjecture.
Conjecture 3.4 Any class e2i of even index is non-trivial in the rational cohomology of the Torelli group Ig for sifficiently large g . Moreover the Sp–
invariant part of the rational cohomology of Ig stabilizes and we have an isomorphism
lim H ∗ (Ig ; Q)Sp ∼
= Q[e2 , e4 , · · · ].
g→∞
At present, even the non-triviality of the first one e2 is not known. One of
the difficulties in proving this lies in the fact that the rational cohomology of
Ig is infinite dimensional in general. Mess observed this fact for g = 2, 3 and
recently Akita [1] proved that H ∗ (Ig ; Q) is infinite dimensional for all g ≥ 7.
His argument can be roughly described as follows. He compares the orbifold
Euler characteristic of Mg given by Harer–Zagier in [52] with that of Ag given
by Harder [45] to conclude that the Euler number of Tg , if defined, cannot
be an integer because the latter number is much larger than the former one.
On the other hand, it seems to be extremely difficult to construct a family of
Riemann surfaces such that its monodromy does not act on the homology of the
fiber wheras the moduli moves in such a way that the classes e2i are non-trivial
(see a recent result of I Smith described in [2] for example). Perhaps completely
different approaches to this problem along the lines of works of Jekel [59] or
Klein [82] might also be possible.
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Shigeyuki Morita
Symplectic representation theory
As was explained in section 2, it is an important method of studying the structure of the mapping class group to combine those of the Siegel modular group
Sp(2g, Z) and the Torelli group Ig together with the action of the former group
on the structure of the latter group. More precisely, there arise various representations of the algebraic group Sp(2g, Q) in the study of Mg . For example, the rational homology group HQ = H1 (Σg ; Q) of the surface Σg is
the fundamental representation of Sp(2g, Q) and Johnson’s result implies that
H1 (Ig ; Q) ∼
= Λ3 HQ /HQ is also a rational representation of it. Hereafter, the
representation Λ3 HQ /HQ will be denoted by UQ . Thus the classical representation theory of Sp(2g, Q) can play crucial roles.
On the other hand, as was already mentioned in the introduction, Kontsevich
[85, 86] used Weyl’s classical representation theory to describe invariant tensors
of various representation spaces which appear in low dimensional topology in
terms of graphs. In this section, we adopt this method to describe invariant
tensors of various Sp–modules related to the mapping class group as well as
the Torelli group.
As is well known, irreducible representations of Sp(2g, Q) can be described as
follows (see a book by Fulton and Harris [29]). Let sp(2g, C) be the Lie algebra
of Sp(2g, C) and let h be its Cartan subalgebra consisting of diagonal matrices. Choose a system of fundamental weights Li : h→R (i = 1, · · · , g) as in [29].
Then for each g –tuple (a1 , · · · , ag ) of non-negative integers, there exists an irreducible representation with highest weight (a1 + · · ·+ ag )L1 + (a2 + · · ·+ ag )L2 +
· · ·+ag Lg . In [29], this representation is denoted by Γa1 ,··· ,ag . In this paper, following [6] we use the notation [a1 +· · ·+ag , a2 +· · ·+ag , · · · , ag ] for it. In short,
irreducible representations of Sp(2g, C) are indexed by Young diagrams whose
number of rows are less than or equal to g . These representations are all rational representations defined over Q so that we can consider them as irreducible
representations of Sp(2g, Q). For example HQ = Γ1 = [1], UQ = Γ0,0,1 = [111]
(which will be abbreviated by [13 ] and similarly for others with duplications)
and S k HQ = Γk = [k] where S k HQ denotes the k -th symmetric power of HQ .
RecallPfrom section 2 that ω0 ∈ H ⊗2 denotes the symplectic class defined as
ω0 = i (xi ⊗ yi − yi ⊗ xi ) for any symplectic basis x1 , · · · , xg , y1 , · · · , yg of H .
As is well known, ω0 is the generator of (HQ⊗2 )Sp . Also the intersection pairing
µ : H ⊗ H→Q serves as the generator of Hom(HQ⊗2 , Q)Sp .
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Structure of the mapping class groups of surfaces
4.1
Invariant tensors of HQ⊗2k and its dual
It is one of the classical results of Weyl that any invariant tensor of HQ⊗2k ,
namely any element of (HQ⊗2k )Sp can be described as follows. A linear chord
diagram C with 2k vertices is a decomposition of the set of labeled vertices
{1, 2, · · · , 2k − 1, 2k} into pairs {(i1 , j1 ), (i2 , j2 ), · · · , (ik , jk )} such that i1 <
j1 , i2 < j2 , · · · , ik < jk (cf Bar-Natan [10], see also [34]). We connect two
vertices in each pair (is , js ) by an edge so that C becomes a graph with k
edges. We define sgn C by
1 2 · · · 2k − 1 2k
sgn C = sgn
.
i1 j1 · · ·
ik
jk
It is easy to see that there are exactly (2k − 1)!! linear chord diagrams with 2k
vertices. For each linear chord diagram C , let
aC ∈ (HQ⊗2k )Sp
be the invariant tensor defined by permuting the tensor product (ω0 )⊗k in such
a way that the s-th part (ω0 )s goes to (HQ )is ⊗ (HQ )js , where (HQ )i denotes
the i-th component of HQ⊗2k , and multiplied by the factor sgn C . We also
consider the dual element
αC ∈ Hom(HQ⊗2k , Q)Sp
which is defined by applying the intersection pairing µ on each two components
corresponding to pairs (is , js ) of C and multiplied by sgn C . Namely we set
αC (u1 ⊗ · · · ⊗ u2k ) = sgn C
k
Y
uis · ujs
(ui ∈ HQ ).
s=1
Let us write
D ` (2k) = {Ci ; i = 1, · · · , (2k − 1)!!}
for the set of all linear chord diagrams with 2k vertices.
Lemma 4.1 dim(HQ⊗2k )Sp = dim Hom(HQ⊗2k , Q)Sp = (2k − 1)!! for k ≤ g .
Proof Let x1 , · · · , xg , y1 , · · · , yg be a symplectic basis of H . There are 2g
members in this basis while if k ≤ g , then there are only 2k (≤ 2g) positions
in the tensor product HQ2k . It is now a simple matter to construct (2k − 1)!!
elements ξj in HQ2k such that αCi (ξj ) (Ci ∈ D ` (2k)) is the identity matrix.
Hence the elements {αCi }i are linearly independent. By the obvious duality, the
Sp–invariant components of tensors {aCi }i are also linearly independent.
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Shigeyuki Morita
Remark The stable range of the Sp–invariant part of HQ⊗2k , which is k ≤ g ,
is twice the stable range of the irreducible decomposition of it, which is k ≤ g2 .
A similar statement is true for other Sp–modules related to the mapping class
group, eg, Λ∗ (Λ3 HQ ) and Λ∗ UQ (see Remark at the end of section 4.2).
Let C, C 0 ∈ D ` (2k) be two linear chord diagrams with 2k vertices. Then the
number αC (aC 0 ) is given by
αC (aC 0 ) = sgn(C, C 0 )(2g)r
where r is the number of connected components of the graph C ∪ C 0 and
sgn(C, C 0 ) = ±1 is suitably
defined. If k ≤ g , then Lemma 4.1 above implies
that the matrix αCi (aCj ) is non-singular. If we go into the unstable range,
degenerations occur and it seems to be not so easy to analyze them. However,
the first degeneration turns out to be remarkably simple and can be described
as follows.
Proposition 4.2 If g = k − 1, then the dimension of Sp–invariant part of
HQ⊗2k is exactly one less than the stable dimension. Namely
dim(HQ⊗2k )Sp = (2k − 1)!! − 1
and the unique linear relation between the elements aC (C ∈ D ` (2k)) is given
by
X
aC = 0.
C∈D ` (2k)
Sketch of proof For k = 1 the assertion is empty and for k = 2 we can
check the assertion by a direct computation. Using the formula for the number
αC (aC 0 ) given above, it can be shown that
X
αC 0 (aC ) = 2k g(g − 1) · · · (g − k + 1)
C∈D ` (2k)
P
for any C 0 ∈ D ` (2k). Hence
C∈D ` (2k) aC = 0 for g ≤ k − 1. On the other
hand, we can inductively construct (2k − 1)!! − 1 elements in (HQ⊗2k )Sp which
are linearly independent for g = k − 1.
Remark After we had obtained the above Proposition 4.2, a preprint by Mihailovs [101] appeared in which he gives a beautiful basis of (HQ⊗2k )Sp for all
genera g . Members of his basis are linearly ordered and the above element
Geometry and Topology Monographs, Volume 2 (1999)
Structure of the mapping class groups of surfaces
363
P
C∈D ` (2k) aC appears as the last one
ment ω k in his notation is equal to k!
for g = k . (More precisely, his last eletimes our element above.) In particular,
the dimension formula above follows immediately from his result. We expect
that we can use his basis in our approach to the Faber’s conjecture (see section
6.4 for more details).
4.2
Invariant tensors of Λ∗ (Λ3 HQ ) and Λ∗ UQ
In our paper [118], we described invariant tensors of Λ∗ (Λ3 HQ ) and Λ∗ UQ (or
rather those of their duals) in terms of trivalent graphs. It turns out that they
are specific cases of Kontsevich’s general framework given in [85, 86]. Here
we briefly summarize them. These descriptions were utilized in [118, 76] to
construct explicit group cocycles for the characteristic classes e ∈ H 2 (Mg,∗ ; Q)
and ei ∈ H 2i (Mg ; Q) (see section 6.4 for more details).
As is well known, Λ2k (Λ3 HQ ) can be considered as a natural quotient as well as
a subspace of HQ⊗6k . More precisely, let p : HQ⊗6k →Λ2k (Λ3 HQ ) be the natural
projection and let i : Λ2k (Λ3 HQ )→HQ⊗6k be the inclusion induced from the
embedding
X
Λ3 HQ 3 u1 ∧ u2 ∧ u3 7→
sgn σ uσ(1) ⊗ uσ(2) ⊗ uσ(3) ∈ HQ⊗3
σ
Λ2k HQ⊗3
and the similar one
⊂ HQ⊗6k , where σ runs through the symmetric
group S3 of degree 3. Then for each linear chord diagram C ∈ D ` (6k), we
have the corresponding elements
p∗ (aC ) ∈ (Λ2k (Λ3 HQ ))Sp ,
i∗ (αC ) ∈ Hom(Λ2k (Λ3 HQ ), Q)Sp .
Out of each linear chord diagram C ∈ D ` (6k), let us construct a trivalent graph
ΓC having 2k vertices as follows. We group the labeled vertices {1, 2, · · · , 6k}
of C into 2k classes {1, 2, 3}, {4, 5, 6}, · · · , {6k − 2, 6k − 1, 6k} and then join the
three vertices belonging to each class to a single point. This yields a trivalent
graph which we denote by ΓC . It can be easily seen that if two linear chord diagrams C, C 0 yield isomorphic trivalent graphs ΓC , ΓC 0 , then the corresponding
elements coincide
p∗ (aC ) = p∗ (aC 0 ),
i∗ (αC ) = i∗ (αC 0 ).
On the other hand, it is clear that we can lift any trivalent graph Γ with 2k
vertices to a linear chord diagram C such that Γ = ΓC . Hence to any such
trivalent graph Γ , we can associate invariant tensors
aΓ ∈ (Λ2k (Λ3 HQ ))Sp ,
αΓ ∈ Hom(Λ2k (Λ3 HQ ), Q)Sp
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Shigeyuki Morita
by setting aΓ = p∗ (aC ) and αΓ =
Γ.
1
(2k)!
i∗ (αC ) where C ∈ D ` (6k) is any lift of
Now let G2k be the set of isomorphism
classes of connected trivalent graphs
`
with 2k vertices and let G = k≥1 G2k be the disjoint union of G2k for k ≥ 1.
Let Q[aΓ ; Γ ∈ G] be the polynomial algebra generated by the symbol aΓ for
each Γ ∈ G .
Proposition 4.3 The correspondence G2k 3 Γ 7→ aΓ ∈ (Λ2k (Λ3 HQ ))Sp defines a surjective algebra homomorphism
Q[aΓ ; Γ ∈ G]−→(Λ∗ (Λ3 HQ ))Sp
which is an isomorphism in degrees ≤ 2g
3 . Similarly the correspondence G2k 3
Γ 7→ αΓ ∈ Hom(Λ2k (Λ3 HQ ), Q)Sp defines a surjective algebra homomorphism
Q[αΓ ; Γ ∈ G]−→ Hom(Λ∗ (Λ3 HQ ), Q)Sp
which is an isomorphism in degrees ≤
2g
3 .
Next we consider invariant tensors of Λ∗ UQ and its dual. We have a natural
surjection p : Λ3 HQ →UQ and this induces a linear map p∗ : Λ∗ (Λ3 HQ )→Λ∗ UQ .
If a trivalent graph Γ ∈ G2k has a loop, namely an edge whose two endpoints are
0 be the subset of G
the same, then clearly p∗ (aΓ ) = 0. Thus let
G2k
2k consisiting
`
0
0
of those graphs without loops and let G = k G2k . For each element Γ ∈ G 0 , let
bΓ = p∗ (aΓ ). Also let q : Λ3 HQ →Λ3 HQ be the Sp–equivariant linear map de1
fined by q(ξ) = ξ− 2g−2
Cξ∧ω0 (ξ ∈ Λ3 HQ ) where C : Λ3 HQ →HQ is the contraction. Since q(HQ ) = 0, it induces a homomorphism q : UQ →Λ3 HQ and hence
0 , let β : Λ2k U →Q be
q : Λ2k UQ →Λ2k (Λ3 HQ ). Now for each element Γ ∈ G2k
Γ
Q
defined by βΓ = αΓ ◦ q .
0 3 Γ 7→ b ∈ (Λ2k U )Sp defines a
Proposition 4.4 The correspondence G2k
Γ
Q
surjective algebra homomorphism
Q[bΓ ; Γ ∈ G 0 ]−→(Λ∗ UQ )Sp
0
which is an isomorphism in degrees ≤ 2g
3 . Similarly the correspondence G2k 3
2k
Sp
Γ 7→ βΓ ∈ Hom(Λ UQ , Q) defines a surjective algebra homomorphism
Q[βΓ ; Γ ∈ G]−→ Hom(Λ∗ UQ , Q)Sp
which is an isomorphism in degrees ≤
2g
3 .
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Structure of the mapping class groups of surfaces
365
Since Λ3 HQ ∼
= UQ ⊕ HQ , there is a natural decomposition
Λ2k (Λ3 HQ ) ∼
= Λ2k UQ ⊕ (Λ2k−1 UQ ⊗ HQ ) ⊕ · · · ⊕ (UQ ⊗ Λ2k−1 HQ ) ⊕ Λ2k HQ
and it induces that of the corresponding Sp–invariant parts. Hence we can also
decompose the space of invariant tensors of Λ∗ (Λ3 HQ ) and its dual according to
the above splitting. In fact, Proposition 4.4 gives the Λ∗ UQ –part of Proposition
4.3. We can give formulas for other parts of the above decomposition which are
described in terms of numbers of loops of trivalent graphs. We refer to [77] for
details.
Remark As is described in the above propositions, the stable range of the Sp–
invariant part of Λ2k (Λ3 HQ ) and Λ2k UQ is 2k ≤ 2g
3 . This range coincides with
Harer’s improved stability range of the homology of the mapping class group
given in [50]. It turns out that this is far more than just an accident. In fact,
this fact will play an essential role in our approach to the Faber’s conjecture
(see section 6.4 and [120] for details).
4.3
Invariant tensors of hg,1
In this subsection, we fix a genus g and we write Lg,1 = ⊕k Lg,1 (k) for the free
Lie algebra generated by H . Also we consider the module
hg,1 (k) = Ker(H ⊗ Lg,1 (k + 1)→Lg,1 (k + 2))
which is the degree k summand of the Lie algebra consisting of derivations of
Lg,1 which kill the symplectic class ω0 ∈ Lg,1 (2) (see the next section section 5
Q
for details). We simply write LQ
g,1 and hg,1 (k) for Lg,1 (k) ⊗ Q and hg,1 (k) ⊗ Q
respectively. We show that invariant tensors of hQ
g,1 (2k) or its dual, namely
Q
Q
Sp
Sp
any element of hg,1 (2k) or Hom(hg,1 (2k), Q) can be represented by a linear
combination of chord diagrams with (2k + 2) vertices. Here a chord diagram
with 2k vertices is a partition of 2k vertices lying on a circle into k pairs where
each pair is connected by a chord. Chord diagrams already appeared in the
theory of Vassiliev knot invariants (see [10]) and they played an important role.
In the following, we will see that they can play another important role also in
our theory.
To show this, we recall a well known characterization of elements of LQ
g,1 (k)
⊗k
in HQ . There are several such characterizations which are given in terms of
various projections HC⊗k →Lg,1 (k) ⊗ C (see [135]). Here we adopt the following
one.
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Shigeyuki Morita
Lemma 4.5 Let Sk be the symmetric group of degree k and let σi = (12 · · · i)
∈ Sk be the cyclic permutation. Let pk = (1−σk )(1−σk−1 ) · · · (1−σ2 ) ∈ Z[Sk ]
which acts linearly on HQ⊗k . Then p2k = kpk and an element ξ ∈ HQ⊗k belongs
Q
to LQ
g,1 (k) if and only if pk (ξ) = kξ . Moreover Lg,1 (k) = Im pk .
⊗(k+1)
If we consider LQ
g,1 (k + 1) as a subspace of HQ
, then the bracket operation
Q
H Q ⊗ LQ
g,1 (k + 1)−→Lg,1 (k + 2)
is simply given by the correspondence u ⊗ ξ 7→ u ⊗ ξ − ξ ⊗ u (u ∈ HQ , ξ ∈
Q
LQ
g,1 (k + 1)). Hence it is easy to deduce the following characterization of hg,1 (k)
⊗(k+2)
inside HQ
.
⊗(k+2)
⊗(k+2)
Proposition 4.6 An element ξ ∈ HQ
belongs to hQ
if
g,1 (k) ⊂ HQ
and only if the following two conditions are satisfied. (i) (1 ⊗ pk+1 )ξ = (k + 1)ξ
and (ii) σk+2 ξ = ξ .
Sp as follows. Recall that we
We can construct a basis of (HQ ⊗ LQ
g,1 (2k + 1))
`
write D (2k) for the set of linear chord diagrams with 2k vertices so that it
gives a basis of (HQ⊗2k )Sp for k ≤ g (see Lemma 4.1). By Lemma 4.5, we have
H Q ⊗ LQ
g,1 (2k + 1) = Im(1 ⊗ p2k+1 )
where we consider 1⊗ p2k+1 as an endomorphism of HQ ⊗ (HQ ⊗ HQ⊗2k ). Let C0
be the edge which connects the first two of the (2k + 2) vertices corresponding
to HQ ⊗ (HQ ⊗ HQ⊗2k ). For each element C ∈ D ` (2k), consider the disjoint
`
e = C0 C which is a linear chord diagram with (2k + 2) vertices.
union C
Hence we have the corresponding invariant tensor
aCe ∈ (HQ ⊗ (HQ ⊗ HQ⊗2k ))Sp .
Let `C = 1 ⊗ p2k+1 (aCe ). Then by Proposition 4.6, `C is an element of (HQ ⊗
Sp
LQ
g,1 (2k + 1)) .
Proposition 4.7 If k ≤ g , then the set of elements {`C ; C ∈ D ` (2k)} forms
a basis of invariant tensors of HQ ⊗ LQ
g,1 (2k + 1). In particular
Sp
dim(HQ ⊗ LQ
= (2k − 1)!!
g,1 (2k + 1))
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Structure of the mapping class groups of surfaces
Sketch of Proof It can be shown that the elements in {`C ; C ∈ D ` (2k)}
are linearly independent because if we express `C as a linear combination of
⊗(2k+2) Sp
the standard basis of (HQ
) given in Lemma 4.1, then we find `C =
aCe + other terms. On the other hand, we can show that the projection under
1 ⊗ p2k+1 of any member of this standard basis can be expressed as a linear
combination of `C .
Now we consider invariant tensors of hQ
g,1 (2k). Associated to any element C ∈
`
Sp
D (2k) we have the corresponding invariant tensor `C ∈ (HQ ⊗ LQ
g,1 (2k + 1)) .
Consider the element
2k+2
X
i
ξC =
σ2k+2
`C .
i=1
By Proposition 4.6 (in particular condition (ii)), ξC belongs to hQ
g,1 (2k) and it
is clear from the above argument that these elements span the whole invariant
Sp
space hQ
g,1 (2k) . More precisely the cyclic group Z/(2k+2) of order 2k+2 acts
Sp and hQ (2k)Sp is nothing but the invariant
naturally on (HQ ⊗ LQ
g,1 (2k + 1))
g,1
subspace of this action. Although there does not seem to exist any simple
formula for the dimension of this invariant subspace, this procedure gives a
method of enumerating all the elements of it.
Here is another approach to this problem which might be practically better than
the above, in particular in the dual setting. We simply use two conditions (i),
(ii) in Proposition 4.6 in the opposite way. Namely we first consider condition
(ii). Recall that any linear chord diagram C with (2k + 2) vertices gives rise
to an Sp–invariant map
⊗(2k+2)
αC : HQ
−→Q.
i
Let us write Ci for σ2k+2
C (i = 1, · · · , 2k + 2). Then, in view of condition (ii)
⊗(2k+2)
above, the restrictions of αCi to the subspace hQ
are equal
g,1 (2k) ⊂ HQ
to each other for all i. This means that, instead of linear chord diagram C ,
we may assume that all of the vertices of C are arranged on a circle. But then
we obtain a usual chord diagram. Thus we can say that any chord diagram C
Sp
with (2k + 2) vertices defines an element of Hom(hQ
g,1 (2k), Q) . In the case of
Vassiliev knot invariant, the linear space spanned by chord diagrams with 2k
vertices, which is denoted by Gk D c in [10], modulo the (4T) relation serves as
the set of Vassiliev invariants of order k . In our case, the linear space spanned
by chord diagrams with (2k + 2) vertices, namely Gk+1 D c , modulo the relations
Sp (and
coming from condition (i) above can be identified with Hom(hQ
g,1 (2k), Q)
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Shigeyuki Morita
also its dual). In particular, we have a surjection
Sp
Gk+1 D c −→ Hom(hQ
g,1 (2k), Q) .
In section 6.2, we will show experimental results which have been obtained by
explicit computations applying this method.
5
Graded Lie algebras related to the mapping class
group
In this section, we introduce various graded Lie algebras which are related to
the mapping class group. We begin by recalling the definition of the Johnson
homomorphisms briefly (see [63, 115, 118] for details). Let Mg,1 be the mapping
class group of Σg relative to an embedded disk D 2 ⊂ Σg and let Mg,∗ be the
mapping class group of Σg relative to the base point ∗ ∈ D 2 . We write Σ0g
for Σg \ Int D 2 so that π1 Σ0g is a free group of rank 2g . Let ζ ∈ π1 Σ0g be the
element represented by a simple closed curve on Σ0g which is parallel to the
boundary. Then as is well known, we have natural isomorphisms due originally
to Nielsen
Mg ∼
= Out+ π1 Σg , Mg,∗ ∼
= Aut+ π1 Σg
0
∼
Mg,1 = {ϕ ∈ Aut π1 Σ ; ϕ(ζ) = ζ}.
g
By virtue of this, any filtration on the fundamental groups of surfaces induces
those of the corresponding mapping class groups. In particular, the lower central
series induces natural filtrations. More precisely, let Γk (G) denote the k -th
term in the lower central series of a group G, where Γ0 (G) = G and Γk (G) =
[G, Γk−1 (G)] for k ≥ 1. We set
Mg,1 (k) = {ϕ ∈ Mg,1 ; ϕ(γ)γ −1 ∈ Γk (π1 Σ0g ) for any γ ∈ π1 Σ0g }
Mg,∗ (k) = {ϕ ∈ Mg,∗ ; ϕ(γ)γ −1 ∈ Γk (π1 Σg ) for any γ ∈ π1 Σg }
and
Mg (k) = π(Mg,∗ (k))
where π : Mg,∗ →Mg is the natural projection. Thus we obtain a natural filtration {M(k)}k on each of the three types of mapping class groups. It is
easy to see that the first one M(1) is nothing but the Torelli group, namely
Ig,1 , Ig,∗ or Ig . Now we can say that the Johnson homomorphism is the one
which describes the associated graded quotients of the mapping class groups,
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Structure of the mapping class groups of surfaces
with respect to these fitrations, explicitly in terms of derivations of graded Lie
algebras associated to the lower central series of the fundamental groups of
surfaces.
Let us simply write H for H1 (Σg ; Z) as before and let Lg,1 = ⊕k Lg,1 (k) be
the free graded Lie algebra generated by H . As is well known, Lg,1 is the
graded Lie algebra associated to the lower central series of π1 Σ0g , namely we
have natural isomorphisms Lg,1 (k) ∼
= Γk−1 (π1 Σ0g )/Γk (π1 Σ0g ) (see [95]). Let
2
ω0 ∈ Λ H = Lg,1 (2) be the symplectic class and let I = ⊕k≥2 Ik be the ideal of
Lg,1 generated by ω0 . Then a result of Labute [87] says that the quotient Lie
algebra Lg = Lg,1 /I serves as the graded Lie algebra associated to the lower
central series of π1 Σg . Now we define
hg,1 (k) = Ker(H ⊗ Lg,1 (k + 1)→Lg,1 (k + 2))
∼
= Ker(Hom(H, Lg,1 (k + 1))→Lg,1 (k + 2))
hg,∗ (k) = Ker(H ⊗ Lg (k + 1)→Lg (k + 2))
∼
= Ker(Hom(H, Lg (k + 1))→Lg (k + 2))
where the second isomorphisms in each of the terms above are induced by the
Poincaré duality H ∗ ∼
= H . In our previous papers [115, 118], Lg,1 , Lg have
0
been denoted by L , L and also hg,1 (k), hg,∗ (k) have been denoted by Hk0 , Hk ,
respectively.
Then the k -th Johnson homomorphisms
τg,1 (k) : Mg,1 (k)−→hg,1 (k)
τg,∗ (k) : Mg,∗ (k)−→hg,∗ (k)
are defined by the correspondence
Mg,1 (k) 3 ϕ 7→ τg,1 (k)(ϕ) = {[γ] 7→ [ϕ(γ)γ −1 ]} ∈ Hom(H, Lg,1 (k + 1))
(γ ∈ π1 Σ0g ) for Mg,1 and similarly for Mg,∗ . Here [γ] ∈ H denotes the
homology class of γ ∈ π1 Σ0g and [ϕ(γ)γ −1 ] denotes the class of ϕ(γ)γ −1 ∈
Γk (π1 Σ0g ) in Lg,1 (k + 1) = Γk (π1 Σ0g )/Γk+1 (π1 Σ0g ). It can be shown that
Ker τg,1 (k) = Mg,1 (k + 1),
Ker τg,∗ (k) = Mg,∗ (k + 1)
so that we have isomorphisms
Mg,1 (k)/Mg,1 (k + 1) ∼
= Im τg,1 (k) ⊂ hg,1 (k)
Mg,∗ (k)/Mg,∗ (k + 1) ∼
= Im τg,∗ (k) ⊂ hg,∗ (k).
Next we consider the filtration {Mg (k)}k of the usual mapping class group
Mg . As is mentioned above, Mg (k) = π(Mg,∗ (k)) and it was proved in [5]
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Shigeyuki Morita
that π1 Σg ∩ Mg,∗ (k) = Γk−1 (π1 Σg ). Hence we have a natural injection Lg (k) ⊂
hg,∗ (k). We define the k -th Johnson homomorphism
τg (k) : Mg (k)−→hg (k)
by setting τg (k)(ϕ) = τg,∗ (k)(ϕ̃) mod Lg (k) (ϕ ∈ Mg (k)) where hg (k) =
hg,∗ (k)/Lg (k) and ϕ̃ ∈ Mg,∗ (k) is any lift of ϕ.
Thus we obtain three graded modules
hg,1 =
∞
M
hg,1 (k),
hg,∗ =
k=1
∞
M
hg,∗ (k),
hg =
k=1
∞
M
hg (k)
k=1
and it turns out that they have natural structures of graded Lie algebras over Z.
The relations between these three graded Lie algebras hg,1 , hg,∗ , hg are described
simply by the following two short exact sequences
0−→jg,1 −→hg,1 −→hg,∗ −→0
0−→Lg −→hg,∗ −→hg −→0
where
jg,1 =
∞
M
jg,1 (k),
jg,1 (k) = Ker(Hom(H, Ik+1 )→Ik+2 ))
k=2
and Lg = Lg,1 /I is the graded Lie algebra associated to the lower central series
of π1 Σg as before.
Sometimes it is useful to consider the tensor products with Q of the modules
appearing above. Let us denote them by attaching a superscript Q to the
Q
original Z–form. For example LQ
g,1 = Lg,1 ⊗ Q and hg,1 = hg,1 ⊗ Q which we
Q
already used in section 4. In these terminologies, hQ
g,1 , hg,∗ are nothing but the
Q
graded Lie algebras consisting of derivations of LQ
g,1 , Lg with positive degrees
Q
which kill the symplectic class ω0 and hQ
g is equal to the quotient of hg,∗ by
Q
Q
inner derivations. We omit the degree 0 part hQ
g,1 (0) = hg,∗ (0) = hg (0) =
sp(2g, Q) because it is the Lie algebra of the rational form of Mg,1 /Mg,1 (1) =
Mg,∗ /Mg,∗ (1) = Mg /Mg (1) = Sp(2g, Z).
Now consider projective limits of nilpotent groups
Ng,1 = lim
←− k≥1 Ig,1 /Mg,1 (k),
Ng = lim
←− k≥1 Ig /Mg (k)
Ng,∗ = lim
←− k≥1 Ig,∗ /Mg,∗ (k)
which are associated to the filtrations M(k) on the corresponding mapping
class groups. We can tensor these groups with Q to obtain pronilpotent Lie
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Structure of the mapping class groups of surfaces
Q
Q
groups Ng,1
, Ng,∗
, NgQ . Let us write ng,1 , ng,∗ , ng for their Lie algebras and also
let Gr ng,1 , Gr ng,∗ , Gr ng be their associated graded Lie algebras, respectively.
Then the Johnson homomorphism induces embeddings of graded Lie algebras
Gr ng,1 ⊂ hQ
g,1 ,
Gr ng,∗ ⊂ hQ
g,∗ ,
Gr ng ⊂ hQ
g
(in fact, these embeddings are defined at the level of Z–forms). More precisely,
we can identify n(k) with Im τ Q (k) so that we can write Gr n = Im τ Q ⊂ hQ
for any type of decorations {g, 1}, {g, ∗}, g . It is a very important problem to
idetify these Lie subalgebras inside the Lie algebras of derivations.
The cohomological structure of the first type hQ
g,1 of the above three graded Lie
algebras was investigated by Kontsevich in his celebrated papers [85, 86] where
he considered three types of Lie algebras `g , ag , cg which consist of derivations
of certain Lie, associative, and commutative algebras. In fact our hQ
g,1 is nothing
+
but the Lie subalgebra `g of `g consisting of elements with positive degrees.
There are natural injections hg,1 ⊂ hg+1,1 so that we can make the direct limit
Q
hQ
∞ = lim hg,1
g→∞
which is equal to the positive part `+
∞ of `∞ in Kontsevich’s notation. In section
6.5, we will apply one of Kontsevich’s results in the above cited papers to our hQ
∞
and obtain definitions of certain (co)homology classes of outer automorphism
groups Out Fn of free groups Fn of rank n ≥ 2.
The second graded Lie algebra, which we consider in this paper, is the Torelli
Lie algebra which is, by definition, the Malcev Lie algebra of the Torelli group.
The structure of the Torelli Lie algebra has been extensively studied by Hain
in [39, 41]. Here we summarize his results briefly for later use in section 6 (see
the above papers for details). We write tg,1 , tg,∗ , tg for the Torelli Lie algebras
which correspond to three types of the Torelli groups Ig,1 , Ig,∗ , Ig , respectively
(Hain uses the notation t1g for tg,∗ ).
In the above, we considered certain surjective homomorphisms
Ig,1 −→Ng,1 ,
Ig,∗ −→Ng,∗ ,
Ig −→Ng
from each type of the Torelli groups to a tower of torsion free nilpotent groups.
Roughly speaking, the Malcev completion of the Torelli group (or more generally of any finitely generated group) is defined to be the projective limit of such
homomorphisms. Since any finitely generated torsion free nilpotent group N
can be canonically embedded into its Malcev completion N ⊗ Q which is a Lie
group over Q, the Malcev completion of the Torelli groups can be described by
certain homomorphisms
Ig,1 −→Tg,1 ,
Ig,∗ −→Tg,∗ ,
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Shigeyuki Morita
from the Torelli groups into pronilpotent Lie groups over Q. They are characterized by the universal property that for any homomorphism ρ : I→N from
the Torelli group into a pronilpotent group N , there exists a unique homomorphism ρ̄ : T →N such that ρ = ρ̄ ◦ µ where µ : I→T is the homomorphism
given above (we omit the subscripts). Since any nilpotent Lie group is determined by its Lie algebra, the pronilpotent group T is determined by its Lie
algebra t which is a pronilpotent Lie algebra over Q. These are the definitions of the pronilpotent Lie algebras tg,1 , tg,∗ , tg which we would like to call the
Torelli Lie algebras. Let Gr t = ⊕k t(k) be the graded Lie algebra associated to
the lower central series of t and also let Gr I be the graded Lie algebra (over
Z) associated to the lower central series of the Torelli group I . Then a general
fact about the Malcev completion implies that there is a natural isomorphism
Gr t ∼
= (Gr I) ⊗ Q. In particular, we have an isomorphism
t(k) ∼
= (Γk−1 (I)/Γk (I)) ⊗ Q.
Now by the universal property of the Malcev completion, there is a uniquely
defined homomorphism T →N which induces a morphism t→n. This induces
a homomorphism Gr t→ Gr n of the associated graded Lie algebras. Thus we
obtain homomorphisms
tg,1 (k)→ng,1 (k) ⊂ hQ
g,1 (k),
tg,∗ (k)→ng,∗ (k) ⊂ hQ
g,∗ (k),
tg (k)→ng (k) ⊂ hQ
g (k).
In fact, Johnson already observed in [63] that the (k − 1)-th term Γk−1 (Ig,1 ) of
the lower central series of Ig,1 is contained in Mg,1 (k) so that there is a natural
homomorphism
Γk−1 (Ig,1 )/Γk (Ig,1 )−→Mg,1 (k)/Mg,1 (k + 1) ⊂ hg,1 (k)
which is defined over Z.
The third (and the final) Lie algebra, denoted by ug , is the one introduced by
Hain in [39]. It lies, in some sense, between the Torelli Lie algebra t and n. We
have the following commutative diagram
1 −−−→
Ig


y
−−−→ Mg −−−→ Sp(2g, Z) −−−→ 1




y
y
(6)
1 −−−→ Ng ⊗ Q −−−→ Gg −−−→ Sp(2g, Q) −−−→ 1
where Gg is the Zariski closure of the image of lim
←− Mg /Mg (k) in the automorphism group of the Malcev Lie algebra of π1 Σg . Hain applied the relative
Malcev completion (see [42]), which is due to Deligne, to the mapping class
Geometry and Topology Monographs, Volume 2 (1999)
Structure of the mapping class groups of surfaces
373
group Mg relative to the classical representation Mg →Sp(2g, Q). Roughly
speaking, it is the projective limit of representations like in (6) where Gg is replaced by an algebraic group defined over Q such that the image of Mg there
is Zariski dense and Ng ⊗ Q is replaced by a unipotent subgroup. It has the
form
1−→Ug −→Eg −→Sp(2g, Q)−→1
where Eg is a proalgebraic group and Ug is a prounipotent group. Ug is called
the prounipotent radical of the relative Malcev completion of the mapping class
group. Let ug be the Lie algebra of Ug . There are also decorated versions of
these groups. By the definition, we have canonical homomorphisms
Tg −→Ug −→N ⊗ Q
and their Lie algebra version
tg −→ug −→ng .
Now we can state two fundamental results of Hain as follows. The first one
gives a complete description of the relation between the Lie algebras t and u.
Theorem 5.1 (Hain [39]) Assume that g ≥ 3. Then the natural homomorphisms Tg,1 →Ug,1 , Tg,∗ →Ug,∗ , Tg →Ug are all surjective and the kernel of each
is a central subgroup isomorphic to Q.
Equivalent statement can be given for the associated Lie algebras t, u. The
second result is the explicit presentation of the Torelli Lie algebra t. Let U =
Λ3 H/H and let UQ be U ⊗ Q as before. Then the Johnson homomorphism
τgQ (1) : Ig −→UQ
induces a homomorphism between the second cohomology groups
τgQ (1)∗ : Λ2 UQ −→H 2 (Ig ; Q)
and Hain determined the kernel of this homomorphism as follows.
Proposition 5.2 (Hain [41]) Let τgQ (1) : Ig −→UQ be as above. Then
Ker τgQ (1)∗ = [22 ] + [0] ⊂ Λ2 UQ = [16 ] + [14 ] + [12 ] + [22 12 ] + [22 ] + [0]
where the decomposition of Λ2 UQ is valid for g ≥ 6.
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The fact that [22 ] + [0] is contained in the kernel was essentially proved in our
papers [111, 112], though in a more primitive way. In fact, we obtained the
secondary invariant d1 ∈ H 1 (Kg ; Q) (see section 6.6 below for details) from the
vanishing of the trivial summand [0] in H 2 (Ig ; Q) which turned out to be closely
related to the Casson invariant. Hain proved the above result in a systematic
way. In particular, the non-triviality in H 2 (Ig ; Q) of other summands than
[22 ] + [0] was shown by constructing explicit abelian cycles of Ig on which they
take non-zero values.
Theorem 5.3 (Hain [41]) If g ≥ 3, then tg is isomorphic to the completion
of its associated graded Lie algebra Gr tg . Moreover if g ≥ 6, then it has a
presentation
Gr tg = L(UQ )/([16 ] + [14 ] + [12 ] + [22 12 ])
where L(UQ ) denotes the free Lie algebra generated by UQ = [13 ].
Hain obtained similar presentations for other Lie algebras tg,1 , tg,∗ , ug,1 , ug,∗ , ug
as well. Although these presentations are very simple, the proof requires deep
and powerful techniques in Hodge theory. One of the main ingredients is to put
mixed Hodge structures on ug and then on tg which depend on a fixed complex
structure on the reference surface. One important consequence of this is that
after tensoring with C, they are canonically isomorphic to their associated
graded Lie algebras.
Thus for the study of the structure of the mapping class group, it is enough to
investigate the morphisms
tg (k)−→ug (k)−→ Im τgQ (k)
between the three graded Lie algebras Gr tg , Gr ug and ⊕k Im τgQ (k) ⊂ hQ
g .
Another important consequence of Hain’s work is that Im τgQ = ⊕k Im τgQ (k) is
generated by degree one summand Im τgQ (1) = [13 ].
6
A prospect on the structure of the mapping class
group
In this section, we would like to describe some of the various aspects of the
structure of the mapping class group which seem to deserve further investigation
in the future. More precisely, we consider the following topics.
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Structure of the mapping class groups of surfaces
6.1
Structure of the graded Lie algebra hg,1
6.2
Description of the Sp–invariant part of hQ
g,1 (6)
6.3
Kernel of tg → Im τgQ and invariants of 3–manifolds
6.4
Cocycles for the Mumford–Morita–Miller classes
6.5
Cohomology of the graded Lie algebra hQ
∞ and Out Fn
6.6
Secondary characteristic classes of surface bundles
6.7
Bounded cohomology of the mapping class group
6.8
Representations of the mapping class group
6.1
Structure of the graded Lie algebra hg,1
We consider the structure of the graded Lie algebra
hg,1 =
∞
M
hg,1 (k)
k=1
(see section 5). Recall that this is the graded Lie algebra consisting of derivations, with positive degrees, of the free graded Lie algebra Lg,1 = ⊕k Lg,1 (k)
which kill the symplectic class ω0 ∈ Lg,1 (2) = Λ2 H . We omit the degree 0 part
hg,1 (0) ⊗ Q which is isomorphic to sp(2g, Q).
As was explained in section 5, this Lie algebra serves as the target of the Johnson
homomorphisms
τg,1 (k) : Mg,1 (k)/Mg,1 (k + 1)−→hg,1 (k)
so that the main problem is to describe the image
Im τg,1 ⊂ hg,1
as a Lie subalgebra of hg,1 . It is one of the basic results of Johnson that
Im τg,1 (1) = hg,1 (1) = Λ3 H.
Here we mention two important topological applications obtained by Hain in
Q
his fundamental work in [41]. One is that the graded Lie algebra Im τg,1
(the
superscript Q will mean that we take the operation ⊗Q as before) is genQ
erated by the degree one summand Im τg,1
(1) = Λ3 HQ , which was already
mentioned in section 5. The other is that the natural map Mg,1 (k)/Mg,1 (k +
1)→Mg,∗ (k)/Mg,∗ (k + 1) is surjective after tensoring with Q (and hence is an
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Shigeyuki Morita
isomorphism for k 6= 2). As is mentioned in [41], this is an answer to a problem
raised by Asada–Nakamura in [6] where they pointed out the possibility that
Q
jQ
g,1 (4) = [2] might be outside of Im τg,1 (4). This possibility is now settled by
Hain to be true as follows. It is easy to see that jg,1 (2) = Z and it is contained
in Im τg,1 (2). However this is the whole of the intersection of Im τg,1 with the
ideal jg,1 . Namely we have jg,1 (k) ∩ Im τg,1 (k) = {0} for all k ≥ 3. Note that
Im τg,∗ contains Lg as a natural Lie subalgebra which corresponds to the inner
derivations of Lg , as was already mentioned in section 5.
In our paper [115], we introduced certain Sp–equivariant mappings
Tr(2k + 1) : hg,1 (2k + 1)−→S 2k+1 H
(k = 1, 2, · · · )
such that they are surjective (after tensoring with Q) and vanish on Im τg,1 as
well as on [hg,1 , hg,1 ]. We can also show that it vanishes on the ideal jg,1 . Thus
the traces descend to Sp–equivariant mappings
Tr(2k + 1) : hg (2k + 1)−→S 2k+1 H
(k = 1, 2, · · · ).
These mappings are called traces because they were defined using the trace of
some matrix representation of hg,1 (2k + 1).
We can summarize the above as follows. The ideal jg,1 and the image Im τg,1 of
the Johnson homomorphism are almost disjoint from each other and both are
contained in the kernel of the traces. Thus the next problem is to determine
how large is the remaining part of the Lie algebra hg,1 .
After we had found the traces, Nakamura [126] discovered another obstruction
to the surjectivity of τg,1 which has its origin in number theory. More precisely,
it is related to theories of outer Galois representations of the absolute Galois
group over Q initiated by Grothendieck, Ihara and Deligne and developed by
many people (see [38, 57, 18, 19, 127] and references in them). Roughly speaking, Nakamura proved that certain graded Lie algebra, which arises in the above
theories, appears in the cokernel of Johnson homomorphisms. In particular, he
concluded that the dimension of the cokernel of the Johnson homomorphism
Q
τg,∗
: (Mg,∗ (2k)/Mg,∗ (2k + 1)) ⊗ Q−→hQ
g,∗ (2k)
is greater than or equal to a number rk which is the free Z` –rank of a certain
Galois group Gal(Q(k + 1)/Q(k)) associated to the outer representation of
Gal(Q/Q) on some nilpotent quotient of the pro-` fundamental group of P1 \
{0, 1, ∞} (see [126] for details). Thus it may depend on the prime `. However
it is conjectured that rk is the dimension of the degree k part of the free graded
Lie algebra over Q which is generated by one free generator σk of degree k for
each odd k > 1 (Deligne’s motivic conjecture). As Nakamura mentions in his
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Structure of the mapping class groups of surfaces
paper cited above, the appearance of these Galois obstructions, in its primitive
form, was predicted by Takayuki Oda and some closely related work was done
by M. Matsumoto. Certain estimates of these numbers have been obtained by
Ihara [56], M. Matsumoto [96], Tsunogai [141] and others.
Next we consider the images of the Johnson homomorphisms
τg (k) : Mg (k)/Mg (k + 1)→hg (k)
for low degrees. The following results have been obtained.
Im τgQ (1) = [13 ]
Im τgQ (2)
Im τgQ (3)
2
= [2 ]
2
= [31 ]
([61])
([111, 41])
([6, 41])
Nakamura made a list of irreducible decompositions of the Sp–modules hQ
g,1 (k)
for small k by a computer calculation. Utilizing it, we made rather long explicit
Q
Q
computations of the Lie bracket Im τg,1
(1) ⊗ Im τg,1
(3)→hQ
g,1 (4), in the framework of symplectic representation theory, and obtained the following result.
Proposition 6.1 We have
Im τgQ (4) = [42] + [313 ] + [23 ] + [31] + [2]
(g ≥ 4)
Q
and the cokernel of the homomorphism τg,1
(4) : (Mg,1 (4)/Mg,1 (5))⊗Q→hQ
g,1 (4)
Q
2
is isomorphic to [2] + [21 ] where [2] = jg,1 (4).
The following list describes the structure of hQ
g,1 (k) for degrees k ≤ 4.
k
1
2
3
4
jQ
g,1 (k)
[0]
[2]
LQ
g (k)
Im τgQ (k)
Cok τgQ (k) \ Tr
[1]
[12 ]
[21]
[31][212 ][2]
[13 ]
[22 ]
[312 ]
3
[42][31 ][23 ][31][2]
[212 ]
Tr
[3]
Here the term Cok τgQ (k) \ Tr means that we exclude the trace component
from the cokernel of the homomorphism τgQ (k). Thus the summand [212 ] in
Q
hQ
g,1 (4) is not contained in Im τg (4) and it should be considered as a new type
Q
of obstruction for the surjectivity of τg,1
other than the ideal jg,1 , the traces
and the Golois obstructions. In the next degree 5, we found yet another new
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Shigeyuki Morita
obstruction. More precisely, at least one copy of [13 ] ⊂ hQ
g,1 (5) cannot be hit
Q
by τg,1 (5) and this leads us to the non-triviality of H4 (Out F4 ; Q) as will be
shown in section 6.5.
Thus the cokernels of the Johnson homomorphisms seem to grow rapidly according as the degree increases and it will require many more studies before we
Q
can figure out the precise form of Im τg,1
inside hQ
g,1 . We mention that KontQ
sevich gave the irreducible decomposition of hg,1 as an Sp–module in [85, 86]
Q
and it would be desirable to identify Im τg,1
as an explicit Sp–submodule of it.
6.2
Description of Sp–invariant part of hQ
g,1 (6)
Sp of the degree 6
In this subsection, we describe the Sp–invariant part hQ
g,1 (6)
summand of the graded Lie algebra hQ
g,1 to illustrate our method. Here we only
give an outline of our computation rather than the details.
We have an exact sequence
0−→hg,1 (6)−→H ⊗ Lg,1 (7)−→Lg,1 (8)−→0.
Sp = 15 and it can be
By Proposition 4.7, we know that dim(HQ ⊗ LQ
g,1 (7))
Sp = 10. Hence dim hQ (6)Sp = 5 in the stable range.
shown that dim LQ
g,1 (8)
g,1
Sp = 2. Alternatively, we can use the
Also it can be shown that dim jQ
(6)
g,1
method described after Proposition 4.7 as follows. The set of chord diagrams
with 4 chords has 18 elements. If we divide the vector space G4 D c spanned
by it by the (4T) relation, then the dimension reduces to 6 (see [10]). For
our purpose, instead of (4T) relation, we have to put the relation coming from
condition (i) of Proposition 4.6. Then we find by an explicit computation that


(g ≥ 3)
5
Q
Sp
dim hg,1 (6) = 4
(g = 2)


1
(g = 1).
Sp in terms of linWe can also give a basis {αi ; i = 1, · · · , 5} of Hom(hQ
g,1 (6), Q)
ear combinations of chord diagrams. Now we consider the following 5 elements
Sp
{ξi ; i = 1, · · · , 5} in hQ
g,1 (6) . We choose the first two elements ξ1 , ξ2 to be a
Q
2
basis of jg,1 (6)Sp . On the other hand, we know that hQ
g,1 (3) = [21] + [31 ] + [3]
2
Q
where [21] = LQ
g (3), [31 ] = Im τg (3) and [3] is the trace component (see the
table in section 6.1). It turns out that the unique trivial summand [0] in each
Sp under the bracket operation
of Λ2 [21], Λ2 [312 ] and Λ2 [3] survives in hQ
g,1 (6)
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Structure of the mapping class groups of surfaces
Q
Q
Sp of these elΛ2 hQ
g,1 (3)→hg,1 (6). We set ξ3 , ξ4 , ξ5 to be the images in hg,1 (6)
Sp ∼ Q.
ements in the above order. We see that ξ3 is a generator of LQ
=
g (6)
Q
Sp
Clearly ξ4 is contained in Im τg (6) . Also it turns out that we can construct
the first two elements ξ1 , ξ2 by taking brackets of suitable elements from hQ
g,1 (1)
Q
and [3] ⊂ hg,1 (3). This is one of the supporting pieces of evidence for Conjecture 6.10 given in section 6.5 below. Now explicit computation shows that
Sp
det(αi (ξj )) 6= 0. Hence {ξi } forms a basis of hQ
g,1 (6) .
Summing up, we have the following table for the dimensions of Sp–invariant
Q
Q
part of hQ
g,1 (6) corresponding to the ideal jg,1 , inner derivations Lg , Johnson
image Im τgQ and the remaining part Cok τgQ (for g ≥ 3).
Sp
jQ
g,1 (6)
Sp
LQ
g (6)
Im τgQ (6)Sp
Cok τgQ (6)Sp
total
2
1
1
1
5
We can also see from the above table that the dimension of Cok τgQ (6) is exactly
equal to one. Hence it should be equal to Nakamura’s Galois obstruction given
in [126] which is in fact independent of `.
The way of degeneration is interesting also here. More precisely, if we set g = 1,
then it turns out that any of the three elements ξ3 , ξ4 , ξ5 goes to a unique nonSp ∼ Q (up to non-zero scalars). Observe here that if
trivial element in hQ
=
1,1 (6)
g = 1, then the Torelli group is trivial, the ideal j1,1 coincides with the whole of
h1,1 and the traces are all trivial. However the forgetful mapping hg,1 →h1,1 is
not a Lie algebra homomorphism so that there is no contradiction here. It may
be said that, in genus one, topology disappears and only arithmetic remains.
The above is only a special case of degree 6 summand. However we can see
already here a glimpse of some general phenomena. Namely the elements
ξ3 , ξ4 , ξ5 can be defined for all degrees 4k + 2 and turn out to be non-trivial in
Sp
hQ
g,1 (4k + 2) . For example ξ4 is defined to be the image of the Sp–invariant
Q
Q
part of Λ2 [2k + 1, 12 ] in Im τg,1
(4k + 2) where [2k + 1, 12 ] ⊂ Im τg,1
(2k + 1) is
the summand given in [6]. The elements ξ5 for all k > 1 were constructed in
a joint work with Nakamura in which we are trying to understand the Galois
obstructions topologically.
6.3
Kernel of tg → Im τgQ and invariants of 3–manifolds
As is well known, there are close connections between the mapping class group
and 3 dimensional manifolds. More precisely, the Heegaard decomposition of
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Shigeyuki Morita
3–manifolds gives rise to a direct correspondence between the two objects and
the construction of surface bundles over the circle with a given monodromy from
the mapping class group yields another such relation. Also the genus 1 mapping
class group plays a crucial role in the theory of Dehn surgery along framed links
in 3–manifolds, particularly in S 3 by virtue of Kirby’s fundamental result [79].
Hence any invariant of 3–manifolds naturally has certain effect on the structure
of the mapping class group.
Until the discovery of the Casson invariant for homology 3–spheres in 1985, the
Rohlin invariant was almost the unique invariant for 3–manifolds. Motivated
mainly by Witten’s influential work in [144], the theory of topological invariants
of 3–manifolds has been continually and rapidly developing. See [134, 83, 80,
123, 130, 88] as well as their references. It is beyond the scope of this article
to review these developments. In the following, we would like to focus on those
results which have direct relation with the structure of the mapping class group.
If we are given a 3–manifold M together with an embedded oriented surface
Σg ⊂ M , then we can associate a new manifold Mϕ to each element ϕ ∈ Mg
by cutting M along Σg and pasting it back together by ϕ. In the cases where
M is an integral homology sphere and ϕ belongs to the Torelli group Ig , the
resultant manifold Mϕ is again an integral homology sphere. Hence, given any
topological invariant α of homology 3–spheres, with values in a module A,
we can define a mapping α : Ig →A by setting α(ϕ) = α(Mϕ ). Birman and
Craggs [12] first studied such mappings for the case of the Rohlin invariant µ.
Johnson [60] extended their results to obtain a complete enumeration of socalled Birman–Craggs homomorphisms. This result played an important role
in his determination of the abelianization of the Torelli group in [65]. In our
papers [111, 112], we studied the case of the Casson invariant λ and in particular
we obtained an interpretation of λ in terms of the secondary characteristic
classes of surface bundles (see [119] and section 6.6 below). This work has been
generalized in two different ways. One is due to Lescop [89] where she obtained,
among other things, a closed formula which expresses how the Casson–Walker–
Lescop invariant behaves under the cut and paste operation on 3–manifolds.
The other is given by a series of works of Garoufalidis and Levine [32, 33]
where they generalized our results cited above extensively. More precisely, we
considered the effect of Casson invariant on the structure of the Torelli group
Ig while they considered all of the finite type invariants of homology 3–spheres
introduced by Ohtsuki [129, 130]. In particular, they proved that Ohtsuki’s
filtration of the space of homology 3–spheres can be described in terms of
the lower central series of the Torelli group. As a corollary to this statement,
they proved that any primitive type 3k invariant α gives rise to a non-trivial
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Structure of the mapping class groups of surfaces
homomorphism
α : tg (2k)−→Q.
Recall from section 5 that we have a series of homomorphisms
tg (k)−→ug (k)−→ Im τgQ (k)
∼ ug (k) for all k 6= 2 and that tg (2) =
and we know by Hain [39] that tg (k) =
ug (2) + [0]. Thus the main problem concerning the above series is the following.
Problem 6.2 Determine whether the homomorphism tg (k)→ Im τgQ (k) is injective (and hence an isomorphism) for k 6= 2 or not.
The results of Garoufalidis and Levine mentioned above show that the above
problem is crucial also from the point of view of the theory of invariants of 3–
manifolds. We mention that, extending earlier results of Johnson [63], Kitano
[81] proved that the k -th Johnson homomorphism τg,1 (k) : Mg,1 (k)→hg,1 (k)
exactly mesures the higher Massey products of mapping tori which are associated to elements of the subgroup Mg,1 (k). Hence if Problem 6.2 will be affirmatively solved, it would imply that the finite type invariants can be described in
terms of the original Casson invariant together with the Massey products. This
might sound unlikely from the point of view of 3–manifolds invariants. Also
the author learned from J Murakami that the restriction, to the Torelli group,
of the projective representation of Mg associated to the LMO invariant given
in [88, 124] gives rise to a unipotent representation (after suitable truncations).
Hence it should be described by the Torelli Lie algebra tg . This also increases
the importance of Problem 6.2.
Although Hain [41] obtained a presentation of the Torelli Lie algebra tg (see
section 5), it is by no means easy to determine tg (k) (k = 1, 2, · · · ) explicitly
by using it. One way to compute them is to apply Sullivan’s theory [137] of 1–
minimal models to the Torelli group Ig which can be described as follows. First
of all we know by Johnson [61] that tg (1) = UQ = [13 ]. It was a consequence of
the results of [111, 112] that tg (2) contains at least [22 ] + [0] where the trivial
summand [0] reflects the influence of the Casson invariant on the structure of
the Torelli group. We have an isomorphism
tg (2)∗ ∼
= Ker(Λ2 tg (1)∗ →H 2 (Ig )).
As was already mentioned in section 5 (Proposition 5.2), Hain [41] determined
the right hand side to be precisely equal to [22 ] + [0] and he concluded that
tg (2) = [22 ] + [0]
(g ≥ 3).
The next case, namely the case of degree 3 is given by the following result.
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Proposition 6.3 We have isomorphisms
tg (3) ∼
= ug (3) ∼
= Im τg (3) = [312 ] (g ≥ 6).
Sketch of Proof Sullivan’s theory [137] implies that tg (3) can be identified
with the kernel of the following homomorphism
(
!
!)
3
∗
tg (1)∗ ⊗ tg (2)∗
t
(1)
Λ
g
d
Ker
−→H 2 (Ig )
−→
Λ2 tg (2)∗
Λ2 tg (1)∗ ⊗ tg (2)∗
where d ≡ 0 on tg (1)∗ and d : tg (2)∗ →Λ2 tg (1)∗ is the natural injection. The
differential d : Λ2 tg (2)∗ →Λ2 tg (1)∗ ⊗ tg (2)∗ is given by d(α ∧ β) = dα ⊗ β −
dβ ⊗ α (α, β ∈ tg (2)∗ ). It is easy to deduce from this fact that it is injective.
Therefore tg (3)∗ is isomorphic to the kernel of the following mapping
Ker(tg (1)∗ ⊗ tg (2)∗ →Λ3 tg (1)∗ )−→H 2 (Ig )
which is given by the Massey triple product. Hain proved in [41] that all of the
higher Massey products of Ig vanishes for g ≥ 6. Hence passsing to the dual,
we see that tg (3) is determined by the following exact sequence
[ , ]
Λ3 tg (1)−→tg (1) ⊗ tg (2)−→tg (3)−→0.
Here the first mapping is given by
Λ3 tg (1) 3 u ∧ v ∧ w 7→ u ⊗ [v, w] + v ⊗ [w, u] + w ⊗ [u, v]
(u, v, w ∈ tg (1))
and the image of which in tg (3) under the bracket operation is trivial because
of the Jacobi identity. The irreducible decomposition of tg (1) ⊗ tg (2) = [13 ] ⊗
([22 ] + [0]) is given by
tg (1)⊗tg (2) = ([32 1]+[3212 ]+[22 13 ]+[312 ]+[32]+[213 ]+[22 1]+[21]+[13 ])+[13 ].
Explicit computations, corresponding to each summand in the above decomposition, show that the mapping Λ3 tg (1)→tg (1) ⊗ tg (2) hits any summand except
[312 ]. Since we already know that Im τgQ (3) = [312 ], the result follows.
If we inspect the above proof carefully, we find that it was not necessary to use
Hain’s vanishing of Massey triple products of Ig for g ≥ 6. It turns out that the
computation itself contains a proof of the vanishing of them. However for higher
degrees k = 4, 5, · · · , Hain’s result considerably simplifies the computations. By
using this method, we can continue the computation for higher degrees. For
example the degree 4 summand tg (4) can be determined by the following exact
sequence
[ , ]
Λ2 tg (1) ⊗ tg (2)−→ tg (1) ⊗ tg (3) ⊕ Λ2 tg (2)−→tg (4)−→0.
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Here the first mapping is given by
(u ∧ v) ⊗ w 7→ u ⊗ [v, w] − v ⊗ [u, w] − [u, v] ∧ w ∈ tg (1) ⊗ tg (3) ⊕ Λ2 tg (2)
where (u, v ∈ tg (1), w ∈ tg (2)) and the above element vanishes in tg (4) again
by the Jacobi identity. Although our computation is not finished yet, we see no
signs of non-trivial kernel for tg (k)→ Im τgQ (k) (k = 4, 5, 6) so far.
We can also ask how other invariants of homology 3–spheres, for example various Betti numbers of Floer homology [28] or infinitely many homology cobordism invariants the existence of which is guaranteed by a remarkable result of
Furuta [31], will influence the structure of the Torelli group. Also it seems to
be a challenging problem to seek those invariants of homology 3–spheres which
reflect semi-simple informations, rather than the nilpotent ones, of the Torelli
group.
6.4
Cocycles for the Mumford–Morita–Miller classes
In our paper [116], we constructed certain representations ρ1 of the mapping
class groups Mg,∗ , Mg and obtained the following commutative diagram
ρ1
Mg,∗ −−−→


y
1 3
2Λ H
Mg −−−→
1 3
2 Λ H/H
ρ1
o Sp(2g, Z)


y
o Sp(2g, Z).
By making use of a standard fact concerning the cohomology of a group which
is a semi-direct product, we deduced the existence of the following diagram
ρ∗
1
Hom(Λ∗ (Λ3 HQ ), Q)Sp −−−→
H ∗ (Mg,∗ ; Q)
x
x




Hom(Λ∗ UQ , Q)Sp
(7)
−−−→ H ∗ (Mg ; Q).
ρ∗1
If we combine this with Proposition 4.3 and Proposition 4.4 (see section 4.2),
then we obtain the following commutative diagram (which is defined at the
cocycle level)
ρ∗
1
Q[αΓ ; Γ ∈ G] −−−→
H ∗ (Mg,∗ ; Q)
x
x




Q[βΓ ; Γ ∈ G 0 ] −−−→ H ∗ (Mg ; Q).
ρ∗1
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It was proved in [118] that the images of ρ∗1 in (8) contain all of the characteristic classes e, ei (i = 1, 2, · · · ). However the problem of determining whether
the images contain new classes or not remained open. This problem was soon
solved by the introduction of generalized Mumford–Morita–Miller classes due
to Kawazumi in [75]. There is a remarkable work of Looijenga [93] in which
he determined the stable cohomology of Mg with coefficients in any finite dimensional irreducible representation of Sp(2g, Q). In [74], Kawazumi used the
above generalized classes to give a different basis for some of these twisted cohomology groups, thereby adding a topological flavor to Looijenga’s result. We
also mention that Ivanov [58] obtained a stability theorem for twisted cohomology groups of mapping class groups (for a surface with at least one boundary
component).
Let us define R∗ (Mg ) to be the subalgebra of H ∗ (Mg ; Q) generated by the
classes ei . Similarly we define R∗ (Mg,∗ ) to be the subalgebra of H ∗ (Mg,∗ ; Q)
generated by the classes e, ei . We may call them the tautological algebra of the
mapping class groups. These are just the reduction to the rational cohomology of the original tautological algebras R∗ (Mg ), R∗ (Cg ) of the moduli spaces
which are defined to be subalgebras of the individual Chow algebras generated
by the classes κi and c1 (ω) (see [26, 43, 53]).
Theorem 6.4 (Kawazumi–Morita [76, 77]) For any g , the images of ρ∗1 in (8)
coincide with the tautological algebras R∗ (Mg,∗ ) and R∗ (Mg ) of the mapping
class groups.
Thus the homomorphisms ρ∗1 in (7) have rather big kernel. For the lower homomorphism ρ∗1 , Garoufalidis and Nakamura [34] have given an interpretation
of this fact in the framework of symplectic representation theory by showing an
isomorphism
Sp
∼
Λ∗ UQ∗ /([22 ])
= Q[e1 , e2 , · · · ]
which holds in the stable range, where ([22 ]) denotes the ideal of Λ∗ UQ∗ generated by [22 ] ⊂ Λ2 UQ∗ . Their result can be generalized to the case of the upper
ρ∗1 so that we have an isomorphism
Sp
∼
Λ∗ (Λ3 HQ∗ )/([22 ] + [12 ])
= Q[e, e1 , e2 , · · · ]
which holds also in the stable range. Here [12 ] is a certain diagonal summand in
Λ2 (Λ3 HQ∗ ), which has three copies of [12 ], described explicitly in [118]. However
these results are, at present, valid only in the stable range while Theorem 6.4
is true in all degrees. If we pass to the dual context, namely if we consider the
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Structure of the mapping class groups of surfaces
homomorphism
1
(ρ1 )∗ : H∗ (Mg ; Q)−→H∗ ( Λ3 H/H o Sp(2g, Z); Q)
2
induced by ρ1 on homology, then we find that those cycles in (Λ∗ UQ )Sp which
come from the moduli space must have fairly restricted types. In particular,
Faber’s result below (Theorem 6.7) implies that the subspace of (Λ2g−4 UQ )Sp
generated by the moduli cycles is one dimensional and a similar statement is
valid for (Λ2g−2 (Λ3 HQ ))Sp . Hence we can define fundamental cycles
µg,∗ ∈ (Λ2g−2 (Λ3 HQ ))Sp ,
µg ∈ (Λ2g−4 UQ )Sp
which are well defined up to scalars and which should be expressed in terms of
certain linear combinations of Sp–invariant tensors aΓ , bΓ described in section
4.
Starting from basic works on the Chow algebras of the moduli spaces M3 , M4
in [23, 24], Faber made numerous explicit computations concerning the tautological algebra of the moduli spaces. Based on them, he proposed the following
beautiful conjecture about the structure of the tautological algebra R∗ (Mg )
(see [26] for details, in particular for the precise description of part (3) below).
Conjecture 6.5 (Faber [26])
(1) The tautological algebra R∗ (Mg ) of the moduli space Mg “behaves like”
the cohomology algebra of a nonsingular projective variety of dimension
g − 2. More precisely, it vanishes in degrees > g − 2, is one dimensional in
degree g − 2 and the natural pairing Ri (Mg ) × Rg−2−i (Mg )→Rg−2 (Mg )
is perfect. It also satisfies the Hard Lefschetz and the Hodge Positivity
properties with respect to the class κ1 .
(2) The [ g3 ] classes κ1 , · · · , κ[ g ] generate the algebra with no relations in
3
degrees ≤ [ g3 ].
(3) There exist explicit formulas for the proportionalities in degree g − 2.
Several supporting pieces of evidence for this conjecture have been obtained.
Theorem 6.6 (Looijenga [92]) The tautological algebra R∗ (Mg ) is trivial in
degrees > g − 2 and Rg−2 (Mg ) is at most one dimensional. Similarly R∗ (Cg )
is trivial for ∗ > g − 1 and Rg−1 (Cg ) is at most one dimensional.
We mention that Jekel [59] proved the vanishing eg = 0 ∈ H 2g (Mg,∗ ; Q) by
making use of a certain representation Mg,∗ → Homeo+ S 1 (cf [109]). This
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Shigeyuki Morita
gives a purely topological proof of a part of Looijenga’s result above. Using the
results of Mumford in [122] as well as those of Witten [145] and Kontsevich [84]
combined with Looijenga’s theorem above, Faber proved the following.
Theorem 6.7 (Faber [25]) κg−2 is non-zero on Mg so that Rg−2 (Mg ) is one
dimensional.
It follows immediately that Rg−1 (Cg ) is also one dimensional.
The above results are obtained mainly in the framework of algebraic geometry.
Here we would like to describe a topological approach to Faber’s conjecture
which has fairly different feature. Naturally we can obtain relations only in the
rational cohomology algebra rather than the Chow algebra. However we hope
that this approach would have its own meaning.
Our method is very simple. Namely, associated to any unstable relation in
H ∗ (Λ3 HQ )Sp or in H ∗ (UQ )Sp , we can obtain a polynomial relation in the tautological algebra R∗ (Mg,∗ ) or R∗ (Mg ) by applying Theorem 6.4. For example,
we can apply Proposition 4.2 to obtain a series of non-trivial relations as follows.
We know by this proposition that there is a unique relation
X
aC = 0
(9)
C∈D ` (6k)
in (HQ6k )Sp for g = 3k − 1. Fortunately, this relation survives in (Λ2k UQ )Sp
under the natural projection HQ6k →Λ2k UQ . Hence passing to the dual, the
P
2k
relation
C αC = 0 gives rise to a polynomial relation in R (M3k−1 ) which
expresses the class ek as a polynomial in lower ei ’s. It turns out that this
relation is exactly the same as Faber’s relation mentioned in [26] up to a factor
of some powers of 2. The associated generating function appears, in our context,
as a result of enumeration of certain trivalent graphs. One of the merits of our
method is that once we obtain a relation in R∗ (Mg ) for some g , we can obtain
associated relations for all genera < g . This is because of the following reason.
Although the mapping H1 (Σg ; Q)→H1 (Σg−1 ; Q), which is induced by collapsing
the last handle, is not very natural from the point of view of algebraic geometry,
it does induce a natural mapping
⊗2k Sp(2g−2,Q)
(Hg⊗2k )Sp(2g,Q) −→(Hg−1
)
where Hg and Hg−1 stand for H1 (Σg ; Q) and H1 (Σg−1 ; Q) respectively. For
example, the relation (9) which is the unique relation for g = 3k − 1 continues
to hold for all g < 3k − 1. In this way, using the unique relation (9) above, we
can prove the following theorem.
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Theorem 6.8 The tautological algebra R∗ (Mg ) of Mg is generated by the
first [ g3 ] Mumford–Morita–Miller classes
e1 , e2 , · · · , e[ g ] .
3
Moreover there are explicit formulas which express any class ej (j > [ g3 ]) as a
polynomial in the above classes.
This theorem gives an affirmative solution (at the level of rational cohomology)
to a part of Faber’s conjecture (Conjecture 6.5). Details will be given in a
forthcoming paper [120]. We expect that we can obtain further relations in
R∗ (Mg ) as well as in R∗ (Mg,∗ ) by this method. We also expect that the
explicit form of the fundamental cycle µg mentioned above should be closely
related to part (3) of Conjecture 6.5.
Concerning the (co)homology of the moduli space Mg , there is another well
known conjecture due to Witten and Kontsevich (see [84]) which says that
certain natural cycles of Mg constructed by them should be expressed in terms
of the Mumford–Morita–Miller classes. The first case was affirmatively solved
by Penner in [131] and Arbarello and Cornalba made considerable progress on
this problem in [3]. We expect that there would exist certain connection between
their method in the framework of algebraic geometry with our approach given
above which uses symplectic representation theory. Penner [132] described a
related conjecture in the context of his new model of a universal Teichmüller
space.
Also there is a problem concerning the unstable cohomology of Mg or Mg .
Harer and Zagier pointed out in [52] that their determination of the orbifold
Euler characteristic of Mg implies that there must exist many unstable cohomology classes. However it seems that the unstable class constructed by
Looijenga [90] for genus 3 moduli space is the only known explicit example.
Problem 6.9 Construct unstable cohomology classes of Mg explicitly.
6.5
Cohomology of the graded Lie algebra hQ
∞ and Out Fn
Here we consider the (co)homology of the graded Lie algebra
hg,1 =
∞
M
hg,1 (k).
k=1
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First we would like to know the abelianization of hg,1 . The first Johnson homomorphism together with the traces induce a graded Lie algebra homomorphism
M
(τg,1 (1), ⊕k Tr(2k + 1)) : hg,1 −→Λ3 H ⊕
S 2k+1 H
k≥1
which is surjective after tensoring with Q where the target is considered as an
abelian Lie algebra. As in section 5, let us consider the direct limit
h∞ = lim hg,1 ,
g→∞
Q
hQ
∞ = lim hg,1 .
g→∞
It is easy to see that the Johnson homomorphism τg,1 (1) and the traces Tr(2k +
1) are all compatible with respect to the inclusions hg,1 ⊂ hg+1,1 . Namely the
following diagram is commutative
L
2k+1 H
hg,1 −−−→ Λ3 Hg ⊕
g
k≥1 S




y
y
L
2k+1 H
hg+1,1 −−−→ Λ3 Hg+1 ⊕
g+1
k≥1 S
where Hg denotes the module H corresponding to the genus g . In view of
the explicit computations in low degrees we have done so far, it seems to be
reasonable to make the following conjecture.
Conjecture 6.10 The abelianization of the Lie algebra hQ
g,1 is given by τg,1 (1)
and the traces, so that we have an isomorphism
M
Q
Q
Q ∼ 3
H1 (hQ
)
=
h
/[h
,
h
]
Λ
H
⊕
S 2k+1 HQ .
=
Q
g,1
g,1
g,1 g,1
k≥1
A similar statement is true for hQ
∞.
If this conjecture were true, then any element in hQ
g,1 , including the Galois
obstructions, can be described by taking brackets of suitable elements of Λ3 HQ
and S 2k+1 HQ . Regardless of whether the above conjecture is true or not, we
have a homomorphism
M
Hc∗ Λ3 HQ ⊕
S 2k+1 HQ −→Hc∗ (hQ
g,1 )
k≥1
where Hc∗ denotes the continuous cohomology, in the usual sense (cf [85]), of
graded Lie algebras which are infinite dimensional but each degree k summand
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Structure of the mapping class groups of surfaces
is finite dimensional for all k . The Sp–invariant part of the above homomorphism can be written as
M
Sp
Hc∗ Λ3 HQ ⊕
S 2k+1 HQ
k≥1
(10)
Sp
∗
3
∗
∗
3
∗
∗
5
∗
∗
Q
Sp
∼
−→H (h ) .
= Λ (Λ H ) ⊗ Λ (S H ) ⊗ Λ (S H ) ⊗ · · ·
Q
Q
Q
c
∞
Sp denotes the inverse limit of the Sp–invariant part of the conwhere Hc∗ (hQ
∞)
tinuous cohomology of the graded Lie algebras hQ
g,1 which stabilizes. The Lie
Q
Q
algebra hg,1 contains Im τg,1 as a natural Lie subalgebra and by restriction and
passing to the limit, we obtain a series of homomorphisms
H ∗ (hQ )Sp −→H ∗ (Gr n∞ )Sp ∼
= H ∗ (n∞ )Sp ∼
= lim H ∗ (Ng,1 )Sp
c
∞
c
c
g→∞
∗
−→ lim H (Mg,1 ).
(11)
g→∞
Q
Q
Here n∞ = limg→∞ ng,1 (ng,1 (k) ∼
= Im τg,1 (k) ⊂ hg,1 (k)) and the second isomorphism is due to Hain [41] where he showed that ng,1 has a mixed Hodge
structure (depending on a fixed complex structure on the reference surface)
so that there is a canonical isomorphism of ng,1 with Gr ng,1 after tensoring
with C. For the last homomorphism, see [118]. Since Tr(2k + 1) is trivial on
Im τg,1 (2k + 1) for any k (see [115]), the composition of (10) with (11) is trivial
on any Sp–invariant which contains the trace component S 2k+1 HQ∗ . Hence, the
moduli part of the homomorphism (10) is given by
Sp
Λ∗ (Λ3 HQ∗ )Sp −→Hc∗ (hQ
∞) .
As was explained in [43, 76] (see also section 2), a combination of our result
[76] with that of [41] implies that the image of the above homomorphism can
be identified with the polynomial algebra
Q[e1 , e2 , · · · ] ⊂ lim H ∗ (Mg,1 ; Q).
g→∞
Then it is a natural question to ask the geometric meaning of the remaining
part of the homomorphism (10). This can be answered by invoking important
work of Kontsevich [85, 86], which we now briefly review.
As was mentioned in section 5, our hQ
∞ is the degree positive part of his Lie
algebra `∞ described in the above cited papers. The homology group H∗ (`∞ )
has a natural structure of a commutative and cocommutative Hopf algebra,
where the multiplication comes from the sum operation `∞ ⊕ `∞ →`∞ . Also it
can be decomposed as
H∗ (`∞ ) ∼
= H∗ (sp(2∞; Q)) ⊗ H∗ (hQ
∞ )Sp .
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By making use of his theory of graph cohomology together with a result of
Culler–Vogtmann [17], he proved the following remarkable theorem.
Theorem 6.11 (Kontsevich [85, 86]) There exists an isomorphism
M
P Hk (`∞ ) ∼
H 2n−2−k (Out Fn ; Q)
= P Hk (sp(2∞; Q)) ⊕
n≥2
where P Hk denotes the primitive part of the k –dimensional homology and
Out Fn denotes the outer automorphism group of the free group of rank n.
More precisely, for each even degree 2n (n > 0) with respect to the grading of
`∞ , there exists an isomorphism
P Hk (`∞ )2n ∼
= H 2n−k (Out Fn+1 ; Q).
Passing to the dual, we also have an isomorphism
Sp ∼
P Hck (hQ
∞ )2n = H2n−k (Out Fn+1 ; Q).
Let us observe here that if Conjecture 6.10 were true, then H1 (hQ
∞ )Sp = 0
2n−3
so that we can conclude H
(Out Fn ; Q) = 0 for any n ≥ 2 by the above
theorem of Kontsevich. We have checked that this is the case up to n = 4.
We mention that Culler–Vogtmann proved, in the above paper, that the virtual
cohomological dimension of Out Fn is equal to 2n − 3.
Now we apply Theorem 6.11 to the homomorphism (10). We know that there
3
is a copy S 3 HQ = [3] ⊂ hQ
g,1 (3) which goes to S HQ bijectively by the trace
Tr(3).
Proposition 6.12 The homomorphism
Λ2 S 3 HQ 3 ξ ∧ η 7−→ [ξ, η] ∈ hQ
g,1 (6)
(ξ, η ∈ S 3 HQ )
is injective.
Sketch of Proof It is easy to see that the irreducible decomposition of the
module Λ2 S 3 HQ = Λ2 [3] is given by
Λ2 S 3 HQ = [51] + [4] + [32 ] + [22 ] + [12 ] + [0].
Then the result follows from rather long explicit computations, in the framework
of symplectic representation theory, of the Lie bracket Λ2 [3]→hQ
g,1 (6).
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Recall that similar homomorphism
Λ2 (Λ3 HQ )−→hQ
g,1 (2)
has a big kernel (Hain [41], see also [118] for the case of one boundary component). Thus we find that the behaviour of the trace component is fairly
different from that of the moduli part. In particular, the pull back of any class
3
in H 2 (S 3 HQ ) to the subalgebra of hQ
g,1 generated by S HQ is trivial. However
if we consider the interaction of the trace component with the moduli part, the
situation changes. In fact, we obtain the following result.
Proposition 6.13 The pull back of the Sp–invariant part H 2 (S 3 HQ )Sp ∼
=Q
2
Q
Sp
to Hc (h∞ ) by the trace Tr(3) is a non-trivial primitive element.
P
Sketch of Proof By Proposition 6.12, there is a chain u =
i (ai , bi ) ∈
3 H ⊂ hQ (3) and ∂u is a non-zero element of
C2 (hQ
)
such
that
a
,
b
∈
S
i i
Q
g,1
g,1
Sp . On the other hand, explicit computation shows that there is a chain
hQ
(6)
g,1
P
Q
Q
0
0
v = j (a0j , b0j ) ∈ C2 (hQ
g,1 ) such that aj ∈ hg,1 (1), bj ∈ hg,1 (5) and ∂v = ∂u.
We can arrange the above elements so that the 2–cycle u − v is in fact an
Sp
Sp–invariant one. Thus it defines an element of H2 (hQ
∞ )6 on which the cohomology class in question takes a non-zero value. The primitivity follows from
the property of the trace.
If we combine the above result with Thoerem 6.11, we can conclude the nontriviality of H4 (Out F4 ; Q). This seems to be consistent with a recent result of Hatcher–Vogtmann [54] in which they proved that H4 (Aut F4 ; Q) ∼
=Q
(Vogtmann informed us that she obtained an isomorphism H4 (Aut F4 ; Q) ∼
=
H4 (Out F4 ; Q) by a computer calculation).
If we use higher traces Tr(2k+1) in the above consideration, we obtain infinitely
Sp and these in turn give rise to a series of certain
many classes in Hc2 (hQ
∞)
homology classes in
H4k (Out F2k+2 ; Q) (k = 1, 2, · · · ).
It seems highly likely that all of these classes are non-trivial. Also by combining
various trace components as well as the moduli part at the same time, we can
define many primitive (co)homology classes of the Lie algebra `∞ .
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6.6
Shigeyuki Morita
Secondary characteristic classes of surface bundles
Chern classes and Pontrjagin classes are representatives of characteristic classes
of vector bundles and they play fundamental roles in diverse branches of mathematics. We may call these classes primary characteristic classes. In some cases
where these primary classes vanish, there arise various theories of so-called secondary characteristic classes. For example, we have the theory of characteristic
classes of foliations, characteristic classes of flat bundles, the theory of Chern
and Simons and also that of Cheeger and Simons and so on.
In the case of surface bundles, it is natural to call the classes ei the primary
characteristic classes. As was mentioned in section 3, the odd classes e2i−1
come from the Siegel modular group Sp(2g, Z) via the classical representation
ρ0 : Mg →Sp(2g, Z). Hence these classes (with rational coefficients) vanish on
the Torelli group Ig . It is a fundamental question concerning the cohomology
of Ig whether even classes e2i are non-trivial on it or not (see Conjecture 3.4).
On the other hand, it was proved in [118] that any class ei , including the cases
of even i, vanish on the subgroup Kg of the mapping class group Mg , which is
the subgroup generated by all Dehn twists along separating simple closed curves
on Σg (see section 2). The proof of this fact goes roughly as follows. As was
recalled in section 6.4, in our paper [118] we have constructed certain natural
cocycles for ei and by the very definition they vanish on Kg . Thus there are
two reasons with different sources that the odd classes e2i−1 vanish on Kg .
By making use of this fact, we can define the secondary characteristic classes of
surface bundles. One way to do so can be described as follows. For each e2i−1 ,
choose two cocycles c, c0 both of which represent e2i−1 such that c comes from
the Siegel modular group while c0 is a cocycle constructed in [118] using the
linear representation ρ1 described there. Since these two cocycles are cohomologous to each other, there exists a cochain
d ∈ C 4i−3 (Mg ; Q)
such that δd = c − c0 . Now both of c, c0 are 0 on Kg so that if we restrict d to
Kg , it is a cocycle. Hence we can define a cohomology class
di ∈ H 4i−3 (Kg ; Q)
to be the class of the cocycle d ∈ Z 4i−3 (Kg ; Q). It can be shown that the
cohomology class of [d] does not depend on the choices of the cocycles c, c0
modulo the indeterminacy
Im H 4i−3 (Mg ; Q)−→H 4i−3 (Kg ; Q) .
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Structure of the mapping class groups of surfaces
We remark that if Conjecture 3.1 were true, then the above indeterminacy
vanishes, at least in the stable range, so that di would be uniquely defined.
We know that the first one
d1 : Kg −→Q
can be defined uniquely. Here we describe the precise formula for it. Let
τ ∈ Z 2 (Mg ; Z) be Meyer’s signature cocyle given in [100] which is in fact
defined on Sp(2g, Z). It represents − 13 e1 . On the other hand in [110, 114] we
gave another cocycle for e1 . More precisely, in the notation of section 6.4 the
cocycle
1
c0 =
(−3αΓ1 + (2g − 2)αΓ2 )
2g + 1
represents e1 (see [44] for an interpretation of this result as well as others from
the point of view of algebraic geometry). Here Γ1 is the trivalent graph which
has 2 vertices and 2 loops while Γ2 is the trivalent graph with 2 vertices and
without loops (namely the theta graph). The above two cocycles are cohomologous to each other so that there exists a mapping
d1 : Mg −→Q
(g ≥ 2)
such that δd1 = −3τ − c0 . Since Mg is perfect for g ≥ 3 and H1 (M2 ) = Z/10,
the above map d1 is uniquely defined. Also since the restriction of both of τ
and c0 to the subgroup Kg is trivial, we obtain a homomorphism
d1 : Kg −→Q.
This is the definition of our secondary class d1 ∈ H 1 (Kg ; Q).
Theorem 6.14 (Morita [112]) Let ϕ ∈ Kg be a Dehn twist along a separating simple closed curve on Σg such that it divides Σg into two compact surfaces
of genus h and g − h. Then the value of the secondary class d1 ∈ H 1 (Kg ; Q)
on it is given by
12
d1 (ϕ) =
h(g − h).
2g + 1
Moreover d1 is the generator of H 1 (Kg ; Q)Mg ∼
= Q for all g ≥ 2.
In our paper [112], the coefficient of h(g − h) in the above formula for d1 (ϕ)
was not mentioned. However it is easy to deduce it from the results obtained
there. In [119] we gave an interpretation of d1 in terms of Hirzebruch’s signature
defect of certain framed 3–manifolds. Generalizing this, we obtained another
more geometrical definition of higher secondary classes di ∈ H 4i−3 (Kg,1 ; Q).
We expect that these two definitions of di would coincide for all i.
Geometry and Topology Monographs, Volume 2 (1999)
394
Shigeyuki Morita
Remark Hain informed us the following interesting facts. In the above theorem, the number h(g − h) appeared in relation to the Dehn twist ϕ. About the
same time, the same number appeared in the works of Jorgenson [67] and Wentworth [143]. In fact, it plays the role of the principal factor in their asymptotic
formula of Faltings delta function around the divisor ∆h , which is associated
to ϕ, of the Deligne–Mumford compactification Mg . This number plays one
more important role also in a recent work of Moriwaki [121] where he obtained
a certain inequality related to the cone of positive divisors on Mg . In fact, Hain
has an interpretation that these phenomena are more than just a coincidence.
Kawazumi is also trying to develop a theory related to d1 . From the topological point of view, we may say that they are the manifestation of the Casson
invariant in the geometry of the moduli space of curves. We expect that there
should exist very rich structures here which deserve future investigations.
Conjecture 6.15 All of the secondary classes di ∈ H 4i−3 (Kg ; Q) are uniquely
defined and non-trivial if g is sufficiently large. Moreover we have an isomorphism
lim H ∗ (Kg ; Q)Mg ∼
= EQ (d1 , d2 , d3 , · · · ).
g→∞
There has been rapid progress in the theory of the moduli space Mg related to
the primary charactersitic classes κi or ei , namely from the celebrated works
of Witten [145] and Kontsevich [84] to the recent developments reaching to the
Gromov–Witten invariants (see [27, 35] and references in them). In contrast
with this, the secondary classes di seem to be beyond the reach of any explicit
study at present, except for the first one. In relation to the first class d1 , it
seems to be a challenging problem to try to generalize the genus one story
given in Atiyah’s paper [8] to the cases of higher genera in various ways. We
believe that the higher classes di will also eventually play an important role in
hopefully deeper geometrical study of the moduli space.
6.7
Bounded cohomology of Mg
In this subsection we consider the mapping class group from the viewpoint of
Gromov’s bounded cohomology (see [37]). Recall that the bounded cohomology
(with coefficients in R) of a group Γ , denoted by Hb∗ (Γ ), is defined to be
the cohomology of the subcomplex of the ordinary R–valued cochain complex
C ∗ (Γ ; R) consisting of all cochains
c : Γ × · · · × Γ −→R
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Structure of the mapping class groups of surfaces
395
which are bounded as functions. We have a natural homomorphism
Hb∗ (Γ )−→H ∗ (Γ ; R)
and it reflects algebraic as well as geometric properties of the group Γ rather
closely. For example if Γ is amenable, then Hb∗ (Γ ) is trivial in positive degrees
while if Γ is the fundamental group of a closed negatively curved manifold,
then the above map (except for degree one part) is known to be surjective by
an argument due to Thurston (see [37]).
Problem 6.16 Study the bounded cohomology of the mapping class group
Mg . More precisely, determine the kernel as well as the image of the natural
map Hb∗ (Mg )→H ∗ (Mg ; R).
In particular, we may ask whether the characteristic class ei ∈ H 2i (Mg ) can
be represented by a bounded cocycle or not. It was remarked in [109] that
Gromov’s general result in [37] implies that any odd class e2i−1 can be represented by a bounded cocycle. This is because, as mentioned in section 3, these
classes are pull backs of some classical characteristic classes of Sp(2g, Z) which
is a discrete subgroup of Sp(2g, R). It seems to be natural to conjecture that
even classes e2i can also be represented by bounded cocycles. One evidence for
this was given in [109] where we proved that any surface bundle with amenable
monodromy group has trivial characteristic classes as it should be if ei were
all bounded cohomology classes. Meyer’s signature cocycle given in [100] is an
explicit bounded cocycle which represents − 13 e1 (see [111]) but no other explicit
bounded cocycle has been constructed for higher odd classes. We mention here
that the cocycles for ei constructed in [118, 76] using trivalent graphs are far
from being bounded.
Problem 6.17 Construct explicit bounded cocycles of Mg which represent
the characteristic classes ei for i > 1.
The particular case of degree 2, namely the map
Hb2 (Mg )−→H 2 (Mg ; R)
already deserves further investigation. Harer’s determination of H 2 (Mg ) in
[46] together with Meyer’s result [100] mentioned above implies that the above
map is surjective. If g = 1, then M1 = SL(2, Z) so that H 2 (SL(2, Z); R) = 0
while it is well known that Hb2 (SL(2, Z)) is infinite dimensional. If g = 2, then
we know that H 2 (M2 ; R) = 0 because M2 is contractible by a result of Igusa
[55].
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Shigeyuki Morita
Proposition 6.18 The two dimensional bounded cohomology Hb2 (M2 ) of the
genus 2 mapping class group is non-trivial.
Proof We consider the cochain d1 ∈ C 1 (Mg ; Q) described in section 6.6.
If g = 2, then U = 0 so that δd1 = −3τ where τ ∈ Z 2 (M2 ) is Meyer’s
signature 2–cocycle for the genus 2 mapping class group. Since H 1 (M2 ; Q) =
0, d1 is uniquely determined by the above equality. Observe that d1 is not
a bounded cochain because d1 : K2 →Q is a non-trivial homomorphism. We
can now conclude that the bounded cohomology class [τ ] ∈ Hb2 (M2 ) is nontrivial.
Remark Meyer’s signature cocycle has been investigated from various points
of view, see work of Y. Matsumoto [98] for the case of g = 2, Endo [21] and
Morifuji [106] for the cases of hyperelliptic mapping class groups and Morifuji
[105] and Kasagawa [69, 70] for certain geometric aspects of it.
Epstein and Fujiwara [22] proved that the second bounded cohomology of any
Gromov hyperbolic group is infinite dimensional. Although the mapping class
group is not a Gromov hyperbolic group, because it contains many free abelian
subgroup of rather high ranks, it was proved by Tromba [139] and Wolpert
[146] (see also [147]) that the sectional curvature of the Teichmüller space with
respect to the Weil–Petersson metric is negative.
Conjecture 6.19 The mapping Hb2 (Mg )→H 2 (Mg ; R) is not injective for all
g . More strongly, Hb2 (Mg ) would be infinite dimensional.
We recall the following definition which is relevant to the above problem.
Definition 6.20 ([97]). A group Γ is said to be uniformly perfect if there
exists a natural number N such that any element γ ∈ Γ can be expressed as a
product of at most N commutators.
Recall that Mg is known to be perfect for all g ≥ 3 (see [46]).
Conjecture 6.21 The mapping class group Mg is not uniformly perfect for
all g ≥ 3.
It was proved in [97] that if Γ is a uniformly perfect group, then the mapping
Hb2 (Γ )→H 2 (Γ ; R) is injective. Hence if the former part of Conjecture 6.19
were true, then Conjecture 6.21 is also true. In the case of g = 2, M2 is not
perfect. However it is also known that its abelianization is finite, namely we
have H1 (M2 ) ∼
= Z/10. The following result is a companion of Proposition 6.18.
Geometry and Topology Monographs, Volume 2 (1999)
Structure of the mapping class groups of surfaces
397
Proposition 6.22 There is no natural number N such that any element in
the commutator subgroup of the genus 2 mapping class group can be expressed
as a product of at most N commutators.
Proof If we assume the contrary, then it would follow that the value of the
cochain d1 is bounded. But as was mentioned above, this is not the case.
6.8
Representations of the mapping class group
In this subsection, we consider various representations of the mapping class
group. First we mention the Magnus representation of the Torelli group. Let
Mg,1 be the mapping class group of Σg relative to an embedded disk D ⊂ Σg as
before and let Z[Γ ] be the integral group ring of Γ = π1 (Σg \ Int D). Following
a general theory of so-called Magnus representation described in Birman’s book
[11], the author defined in [115] a mapping
ρ : Mg,1 −→GL(2g, Z[Γ ])
and considered various properties of it. It is easy to see that this mapping is
injective. However it is not a homomorphism in the usual sense but is rather
a crossed homomorphism. To obtain a genuine homomorphism, we have to
restrict this mapping to the Torelli group Ig,1 ⊂ Mg,1 and reduce the coefficients to Z[H] which is induced by the abelianization Γ →H . Then we obtain
a homomorphism
ρ : Ig,1 −→GL(2g, Z[H])
(see Corollary 5.4 of [115]).
Problem 6.23 Determine whether the representaion ρ : Ig,1 −→GL(2g, Z[H])
described above is injective or not.
Here we would like to mention that Moody proved in [103] that the Burau
representation of the braid group Bn is not faithful for sufficiently large n
while it seems to be still unknown whether the Gassner representation of the
pure braid group is faithful or not.
The augmentation ideal of Z[H] induces a filtration of Ig,1 . It is easy to see
that this filtration is strictly coarser than {Mg,1 (k)}k≥1 which was described
in section 5. It would be interesting to study how they differ from each other.
Besides the Magnus representation, we have now various representations of the
mapping class group associated to newly developed theories which are related
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Shigeyuki Morita
to low dimensional topology. For example, we have projective representations
of Mg arising from the conformal field thoery (see [144, 140, 83], see also
[148, 30] for certain explicit studies of them) or from the theory of universal
perturbative invariants of 3–manifolds (see [88, 124, 125]). We also have Jones
representations [66] of the hyperelliptic mapping class group and the Prym
representations of certain subgroups of Mg given by Looijenga [94]. It seems
that there are only a few results which clarify how these representations are
related to the structure of the mapping class group.
Problem 6.24 Study various properties of the above representations of the
mapping class group. In particular, determine the kernel as well as the image
of them.
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Department of Mathematical Sciences, University of Tokyo
Komaba, Tokyo 153-8914, Japan
Email: morita@ms.u-tokyo.ac.jp
Received: 30 December 1998
Revised: 29 March 1999
Geometry and Topology Monographs, Volume 2 (1999)